Naturally occurring mind control fungus.

Villains are just normal people who were living their normal lives when they fell victim to a strain of Ophiocordyceps

The fungus takes over the mind of the villain filling them with some 'master plan' or another and driving them to work towards this plan. There is always a fatal flaw in the plan (the fungus doesn't actually want the world to be destroyed after all. It only wants to spread to a new host.) And there is always a moment when a hero and hopefully a number of hostages are captured and loosely restrained.

Eventually, the plan consumes the mind of the villain until it is the ONLY thing on the villain's mind. It is at this point that the monologuing begins. The villain can't help but explain the details of the plan once the fungus has reached this stage.

This is the vital moment for the fungus. The moment when it releases microscopic spores from the mouth of the villain directed towards the restrained hero and hostages, allowing it to spread and reproduce.

Once the monologuing has finished, the villain has served its purpose and is no longer under the influence of the fungus. This is why many villains have a moment of repentance: "What have I done?!"

It is also important that the hero and hostages can be saved and can foil the plan. At least one of those exposed to the spores is likely to become a new host for the fungus and eventually continue the reproductive cycle of the monologuing fungus in the amazing sequel.

Because the deed is already done and the rest is an automaton. Nothing can stop the plan now, so why not gloat a little and inform the self-proclaimed hero that he is too late? But only if there isn't even a need to kill the guy anymore because every competent mastermind knowing his list should just kill them if there might be any reason for that (especially Points 4, 6, 7, 16, 142). Note that a good doomsday device is simple (Point 85) and has no self destruct (Point 9), meaning that even the heroes entering the lair could be the moment the deed is done and our mastermind just starts to elaborate what now happens and nobody can stop it now... Let's ask the greatest of all masterminds how it's done:

My name is Ozymandias, king of kings: Look on my works, ye Mighty, and despair!

eh, wrong Ozymandias. This one:

Alternatively, the countdown of the doomsday clock is a little off and the mastermind ensures his adversary believes he still has time (Point 15)...

Assuming that your super villain isn't from Andromeda IV, where everyone breaths methane, the villain is human. Human beings crave social interaction and attention. Go look at all the stupid bullshit people post on twitter everyday. They are bragging, boasting, and trying to get attention. If Super villains were properly updated to modern times, they would all have social media accounts and would be ranting to the largest audience they could get.

The other aspect about SV to consider is that they are essentially terrorists. A terrorist is someone who is disenfranchised from participating in a political system. A terrorist blows up a bus of civilians, because they have no other route to get attention or enact the change they want other than to commit atrocities. You might have some petty criminals who just want to get rich, or psychopaths who want to kill people for the joy of killing, but most comic-book style SV are just terrorists in spandex.

So once you have a captive audience (haha), of course you will tell them why you are crashing the moon into the earth. The act is an attempt to get attention, and now you have it.

The hero arrived to early

The villian is just waiting for the next stage of his plan to go into place when the hero arrives. To stall for time he brags about the details of the plan so far in order to distract and buy those last few minutes of time.

'Insane Troll Logic' Answer

• Supervillains are called that because they live in super villas.
• Aston Martins are super cars.
• Divide by the numerator, and you get Aston Villa
• Aston Villa was founded in 1874
• 1874 is leet-speak for IBTA, the International Baton Twirling Association
• A baton is the traditionally sign of a field marshal
• Field Marshals always tell their enemies how they will keep to the rules of war
• This is, obviously, why supervillains must always announce their plans to their adversaries.

You could say it's ego. They want to say "hey, look how I outsmarted the hero with this brilliant plan" but in reality it's just a plot device.

Science Related Memetic Disorder (SRMD)

SRMD, often referred to as Mad Scientist's Disease, is a well documented mental disorder. It is memetic, meaning it develops from contagious ideas rather than biological organisms. There are five clearly defined stages that mad scientists pass through:

1. Infection. An idea for world domination occurs to the scientist.
2. Obsession. The scientist works obsessively towards his or her goal.
3. Challenge. The forces of law and order confront the scientist.
4. Chase scene. Self-explanatory.
5. Denouement. The scientist surrenders, sometimes after a brief fight and/or negotiation.

If SRMD is not caught in the early stages, it may not be treatable without completing all five. Without treatment, the cycle will repeat ad nauseam.

As documented in A Miracle of Science, if a mad scientist does not reveal their plot, they cannot complete the cycle and begin recovery. Unfortunately, this is also often how the disease spreads.

The evil plan was not the real goal, beating the hero was.

The villain's goal was never to destroy the world, the villain's goal was to beat the hero (why? Is it his brother? Did his father try to force unattainable hero skills on him? Did his girlfriend leave him for a hero?).

If the hero will simply say "You've beaten me," the villain has won and will stop his plan. But as long as the hero refuses to yield, then the plan must continue to prove he has beaten the hero.

The villain wants the hero to come over to his side. He believes that once he explains the plan, the hero will be so impressed by its genius, that he will join the villain in his quest of evil!

or

The hero has promised that if the villain only explains his plan, he will not interfere with it. The villain, believing that since the hero is "good" he must be honest, takes him at his word.

You're a villain.

You've been planning and plotting and pushing pieces around for literally years behind the scenes.
You've put in the work.
You have minions, but they're well invested in your project, they know the working that you've put in and were onboard from the start. (and anyway they're employees, not comrades)
You've not really had anyone you can talk to about this project you've been dedicating your life to for years.

You can't exactly strike up a conversation with someone in a coffee shop and say "oh yeah, I've been working on my plan to rule the world...let me tell you all about it"
Either they take it seriously, and you've shot yourself in the proverbial foot, or they don't, and while you've been talking they haven't believed you.

So finally, you're in the endgame, and you've got someone "on the outside" in front of you.
Someone to talk to without the pretence or as an employer.
Someone who'll take you seriously.

Your intellectual solitary confinement is over.
Of course you talk about your evil plan!

Since the question is tagged "psychology", here's the proper psychology term for this:

Narcissistic Personality Disorder (NPD)

https://en.wikipedia.org/wiki/Narcissistic_personality_disorder

On top of this, gloating super-villains who literally feel compelled to explain in great details their evil schemes (despite the fact it's super counter productive) may also suffer from:

Obsessive Compulsive Disorder (OCD)

https://en.wikipedia.org/wiki/Obsessive%E2%80%93compulsive_disorder

and/or (mild) Autism Spectrum Condition (ASC)

https://en.wikipedia.org/wiki/Asperger_syndrome

(it's not unusual for people with Asperger's/mild ASC to talk very enthusiastically and in great details about their very specific interests, in a somewhat overwhelming fashion, leaving their audience slightly bemused)

Note: I am only providing pointers for information (and possibly further research). Mental health conditions should be dealt with sensitively in fiction.

In the real world: (from what I understand)

• NPD is rather strongly correlated with abusive behaviour towards others
• ASC is more likely to be at the receiving end of abusive behaviour
• OCD is mostly directed inwards, but the anxiety can "spill out" and turn into aggression

In fiction, I believe (this is subjective...) that gloating supervillains are typically meant to be narcissists. Whereas the rambling, quirky-yet-likable scientist side-sick who comes up with all the heroe's super-gadgets is more reminiscent of ASC.

Something Vimes had learned as a young guard drifted up from memory. If you have to look along the shaft of an arrow from the wrong end, if a man has you entirely at his mercy, then hope like hell that man is an evil man. Because the evil like power, power over people, and they want to see you in fear. They want you to know you're going to die. So they'll talk. They'll gloat.

They'll watch you squirm. They'll put off the moment of murder like another man will put off a good cigar.

So hope like hell your captor is an evil man. A good man will kill you with hardly a word.

― Terry Pratchett, Men at Arms

It is taught in "Evil Boarding School of Villainy for Criminally Insane Children" in at least two subjects every semester. And continued to be taught in "Highschool of Evil" and "College for questionable science and evil experimentation".

"Taught" would be an understatement. Beaten into them would be closer to truth. Because there are only bad people, only some of them are on opposing sides. The villain who is the owner of the school actually doesn't like competition and wants "Heroes" to stop his proteges. He himself never monologues to hero and is the deadliest villain on the planet. https://www.youtube.com/watch?v=BjKpqKLZZAc

And unless you want all heroes and villains to gang up on you because you are unlicensed upstart trying to be supervillain, you will get licensed and certified at one of the institutions mentioned above. Or one of their competitors.

Besides, what's the difference between supervillain and regular villainous scum? Presentation!

https://www.youtube.com/watch?v=k7xy7gFH0lI

And monologuing is part of the "Presentation package" brainwashed into any aspiring villain.

People are social animals. People like to tell stories.

This really isn't far fetched at all. It's gotten a bad rap because it's hackneyed and cliched, not because it's unrealistic. (Unrealistic doesn't seem to bother people anyway; witness people getting blown away by gunshots in the movies.)

Here is a great example. The whole video is well worth a watch, but the relevant part starts at 26:30 or so. Police are exceedingly good at getting confessions out of folks. Just listen to him talk. Hell, the first words out of his mouth elicit a confession from an audience member, after 25 minutes of a lawyer lecturing literally about why you should never talk to the police. He starts off by asking people who drove to the lecture today, and then he asks if anyone maybe drove a little fast to get there. And someone raises their hand!

The police officer goes on to elaborate that getting people to talk is easy, because people hate silence and all you have to do a lot of the time is just sit there, in total silence, and fill out paperwork. A lot of folks will feel compelled to start a conversation. Which is their doom. And they know that, and they do it anyway.

This is addressed in the following way mechanically in the Marvel Super Heroes Role-Playing Game (1984, Campaign Book, p. 33), in the words of Doctor Doom:

While heroes often stand in the path to triumph, they also represent the small portion of humanity that understands and appreciates the achievements of marvelous thinkers like me. It is uplifting and satisfying to explain how I achieved triumph to a person with a mind capable of understanding the fine details of the plan, even if the listener is a bitter foe.

Explaining the vital points of a plan or device to a hero earns a bonus of 20 Karma points. Of course, only fools would do such a thing unless the hero were powerless to interfere.

As Gul Dukat says, "A true victory is to make your enemy see they were wrong to oppose you in the first place. To force them to acknowledge your greatness."

How are they supposed to appreciate how right you are and how wrong they are if they don't know what you're doing in excrutiating detail??

Most of the villains I am aware of became villain as a reaction to the lack of acknowledgement they have experienced at some time during their life.

They seek to rule to world not just because they crave power, but because they crave recognition.

And what is more rewarding that have the non plus ultra of mankind, the brave super hero, stare at you in awe while listening at the wonderful convolutions of your evil plan? Puny weak humans just flee your presence, only super heroes stand your presence and your mind.

Everyone, deep down, believes they are the good guy. Good people want others to acknowledge that they are good, right? Well, if you don't understand why I've rigged these children to a bomb, then I merely look like a bad guy! I want to kill half of all life because it's the right thing. And revenge is reasonable - you've seen the movies, but not knowing how your father stole my father's secret formula makes me look petty. The ends justify the means, but somehow people need to know why the means were justified.

Because it feels so good

Gloating, when you get down to it, is really the ultimate display of superiority. It's essentially telling the hero 'I'm so confident in my plan that I can tell you all of it and there's nothing you can do to stop me!' While I suppose you can take a leaf out of Watchmen and follow up the evil plan gloating with 'Oh, by the way this all happened half an hour ago so you literally can't stop me', that's just not as satisfying. The most satisfying thing is telling the hero in advance and then watching their pathetic, weak, futile struggle against the inevitable success of your evil scheme. Because what’s the point of being a villain if you can't rub it into people's faces and enjoy their salty tears of despair? Is it kind of dumb? Yeah. But the temptation is really there, and villains generally don't get to where they are by ignoring their temptations - generally, the opposite, if anything.