Why do NFTs have value?

They are valuable because people value them. The same way that people like to own other trinkets that have no particular use. In many cases, people like to "own" something, even if that thing can easily be copied.

As an analogy, would you rather have a Van Gogh painting, or a really good copy of a Van Gogh painting? If you don't care then NFTs aren't for you.

Most people would rather have the original painting. NFTs are the way to say "I own the original version of this tweet, all the others are just copies".

Like anything else, the NFT is worth whatever price you can find someone willing to pay for it. Unlike virtually every other asset, the NFT has no other use: it represents ownership for the sake of ownership. Since it's non-fungible, there is no (apparent) substitute that would be equally "useful" to own, so the NFT has value as long as someone else is willing to pay to acquire ownership.

I think the closest precedent are the Rai stones used in the Yap islands for currency. You don't need physical possession of the stone; you only need all relevant parties to acknowledge that you own the stone, regardless of its location.

The other answers so far have missed an important point. As with a lot of expensive art, the object itself is almost beside the point. What's really at work is status seeking. By spending a lot of money on some "art object", even ones that most of us would throw out with the trash, you demonstrate that you:

a) have lots of money

b) have the "taste" necessary to be accepted into the society that values that sort of thing

This is explained in much more detail in the book "The $12 Million Stuffed Shark".

An NFT has no fundamental value. Its selling price is only worth what someone is willing to pay for it. It reminds me of Tulipmania where something worth next to nothing suddenly sells for a fortune.

AFAIC, this is just a new version of pump and dump and I wouldn't be surprised if the bubble pops. That is, assuming the Greater Fool Theory doesn't prevail and affluent investors persist in paying a fortune for the novelty of owning something worthless.

Greater Fool Theory

The greater fool theory argues that prices go up because people are able to sell overpriced securities to a "greater fool," whether or not they are overvalued. That is, of course, until there are no greater fools left.

The idea behind NFTs as art is that they create a kind of scarcity for digital art that already exists and drives the value of physical art. The theory is that the reason high value art has such high value is because of its scarcity and not because of the inherent value of the art.

Let's think about this by way of analogy. Let's suppose an old painting was discovered. It had never been seen before, but some art historians examined the painting and concluded that it could possibly have been made by da Vinci, but more likely it was done by one of his students or someone else entirely. That painting might now have some value. It could potentially be sold at auction for a decent amount of money. Now let's say that forensic research is done on the painting and it's conclusively proven that the artwork was created by da Vinci. Suddenly that exact same artwork is worth a fortune.

In fact, you can ask your exact same question about paintings. Is any painting really with millions of dollars on its own? Do paintings have any inherent value? Why is the identity of the painter relevant to the underlying value of the artwork? The quality is no better than it was before we knew it was a da Vinci. Its state of preservation is no different. All that's different is that we can now know conclusively that it was created by da Vinci. The "inherent value" of that piece of art is whatever it would sell for if it were made by any random anonymous painter. The fact that it's worth a fortune is because of its known provenance coming from da Vinci. NFTs work by establishing the provenance of a digital asset.

Imagine you wanted to buy the "official" Nyan Cat. The way it works is the well-known owner of the work issues an NFT representing Nyan Cat. That NFT is non-fungible (hence the name "non-fungible token") and is always owned by a single individual. It can be transferred or whatever, but it will always be unique. The history of transfers of that token is public and can be traced back to the original creator.

This is what makes NFTs scarce. Can any random individual make an NFT of Nyan Cat? Sure. But it wouldn't be worth anything because it wasn't created by Nyan Cat's known creator. That's like an art student copying da Vinci's work. Could the creator make multiple NFTs? Actually yes. This would reduce the scarcity of the NFT and possibly make it worth less. That would be analogous to a physical artist making multiple copies of their work. Although there would still always be a "first issue" NFT which may be worth more than others.

In the case of the article you mentioned, NFTs representing the artwork are being minted by the owners of the physical paintings. Perhaps that's what makes these NFTs valuable to some buyers despite the artwork being in the public domain and anybody theoretically being able to mint an NFT of it.

Note: This argument only truly applies for extremely notable works of art or artists. I 100% agree with the other answerers who are saying there is currently an NFT mania and tons of junk NFTs are selling for ridiculous amounts of money because of the hype. I'm just pointing out an actual real useful application of NFTs.

Furthermore, there are many other applications of NFT technology that aren't about selling works for enormous sums. For example, you could easily imagine an NFT-based DRM mechanism that only allows the holder of an NFT to access some medium. This could apply to run-of-the-mill music or movie sales at completely normal prices. NFT-based event tickets are already a thing that exists, making the transfer of tickets on secondary markets much more controllable, safer for consumers, and beneficial to event hosts. With some imagination, you can come up with many ways NFT technology can be used to solve real, practical problems.

Without rehashing the idea of bubble mania, my most optimistic answer to "why" would be outside speculation of future abilities to temporarily donate the art to a museum of some sort multiple times.

To some extent some of the value of some art for some buyers is its capacity to generate income tax deductions for the owner. Art can be donated, temporarily, multiple times to museums for use in relevant exhibits.

If I happened to be mining ethereum a few years ago, or bought 100 of them for $100 a handful of years ago, or whatever; breaking off 10 ETH to buy "$25,000" of NFTs (at a cost of $10 to me) on the off chance that I can generate future income tax deductions based on my "$25,000" purchase price (or later appraised value) might be worth it. We may live in a future where someone has figured out how to have museum exhibits including NFTs and assuming similar tax-laws for temporary donations of art, and multiple donations over time, some NFTs could confer tax benefits to the owners in excess of their total acquisition and maintenance costs.

It would probably be more advisable to sell such a volatile asset, book the gain and move on (or better yet, stay away all together). But for true crypto believers or early adopters, the confluence of high volatility, low early entry costs, high flying art appraisals, and income tax benefits related to temporary art donations could sum to a future perfect storm. For the right person the speculative cost of an NFT, or small collection of NFTs, may be worth the gamble.

I think it's pretty obvious that there is some fraud and fraud adjacent activity in NFT markets; which is not too dissimilar from the ICO markets of a couple years ago. Things like undisclosed self dealing. And fraud is a hallmark of a bubble.

If you happened to be a crypto early adopter you’re probably better off just buying a real physical Picasso with your gains rather than an NFT to donate to various exhibits through your life. But, again, if you’re a true believer NFTs may prove to be a good sheep to sheer for the tax deductions in the future.

This would be a US-centric very speculative "why" response.

The other answers here aren't addressing what I believe to be a core misunderstanding within your question.

You said:

So when someone made an NFT of the first Twitter posting,

It wasn't 'someone' that made an NFT of the first Twitter posting. The Twitter founder made an NFT of the very first tweet ever, which was also made by the founder of Twitter. The entire concept around NFT's are tied to who made it, and brushing it off as 'someone' is misunderstanding why some people are attracted to buying them.

If you made an NFT of the first twitter posting, no one would care. It would be like you signing your name on another artists painting.

Now, why do people spend money on NFT's? A mix of reasons. Some is Tulipmania where people are speculating that they will be worth more in the future. Some are interested in the concept of NFT's as a way to support artists and prove patronage.

An NFT owner is effectively an executive producer.

Film and television production has a credit known as "executive producer." Wikipedia's article about this role states that it includes financing the work and may include participating in its production. Especially in television, not every executive producer is a "showrunner" with creative control. As Nobby wrote in an answer about executive producers of America's Got Talent:

These days it is more common for people to 'buy' themselves a credit on TV and film productions through financial backing, sponsorship or other enabling methods.

Traditionally (at least on film), the executive producer(s) coughed up the money to get the production rolling - they generally wouldn't have any other impact on the production.

Someone who purchases a non-fungible token (NFT) from a work's author likewise financially supports the creation of a work by contributing an amount of cryptocurrency in exchange for being publicly credited as the work's NFT owner.

In a few cases, the author agrees to bundle other rights in a work with NFT ownership. This may include a license to commercially exploit a work, as in the case of Kevin Smith's 2021 film Killroy Was Here.

NFTs are not just a way to pay for pretentious art. They simply extend the basic idea of cryptocurrency. Cryptocurrency is a way to prove that you were given certain funds, without having to hold on to paper bills (and having to prove their authenticity). Remember that money itself has no intrinsic value, it is like a part-ownership coupon or an IOU from a central bank. The property of cryptocurrency is that they are all interchangeable aka "fungible". NFTs extend this idea to unique items which are not fungible - so you can prove you own a particular copy of a painting without having to actually have the painting in your vault and demonstrate it to anyone who questions whether you really own it.

NFTs are basically a cryptographically verifiable ownership certificate for individual goods that are not interchangeable with other similar goods. You could have NFTs showing your ownership of your house, your car, your vacuum cleaner (presumably that particular vacuum cleaner with the serial number baked into the NFT), your dog...

You might be realizing that having a certificate does not physically prevent me from violating your ownership. For example, you still have to lock your car, and even if you have the dead I can still hop in and steal it. It is the same with houses and vacuums and dogs. So the ownership certificate (the title in this case) is not a magic incantation, it is merely a formality. But that formality allows you to file a police report and have the state recover your property. Basically, it doesn't matter who you or I think is the rightful owners of things, it matters what the government (ie. bad dudes with guns) thinks because they are the ones, unlike you or me, who will enforce their own opinion on who owns what and what they can do with it.

Currently there is not much legal recognition of NFTs, so you might wonder if they're worthless, since nobody will respect your NFT-based ownership. But they can still be worth something via an expectation that they will be observed in the future. This is also why people buy shares of land on the moon - obviously they can't go to the moon right now, but their thesis is that one day the government will officially open the moon for sale and their claims will be respected. With cryptocurrency, early on many people said that it's worthless because no store accepts them and you can't buy anything with them. Today, you can buy even houses and cars with bitcoin. So those people who bought up some and saved them can now enjoy the benefit of their savings.

The problem with NFTs is that the NFT itself is not unique - unless the underlying good is digital you can make multiple NFTs of the same object. Which NFT is the "correct" one then? Basically, the entire crypto community must agree on some common practice for generating legitimate NFTs and how to judge legitimate ones. When that happens, they become an actual basis for ownership rather than just a potential one. The process can be accelerated by the government blessing a certain NFT mechanism as the "official" one and promising to protect it with force - but this is not necessary if there is critical mass of people voluntarily behaving as if that one NFT system is legitimate. For the present, I believe it NFTs are more valued for:

1. The potential future implications as to the ownership of the object
2. The inherent value of the certificate itself, being a technological novelty (this is similar to how collectible coins or documents can trade far above their face value)

NFTs have value solely because some people agree that they do. The same principle applies to money - bitcoin, dollars, dollar bills, etcetera. It doesn't cost anywhere near $100 to produce a $100 bill.

Unlike a physical good that is priced near the cost of its production, abstractions such as coins, NFTs and the like are not inherently valuable. They are only valuable because some people agree that they are.

Most of economics is just a shared dream/delusion that we have found to be useful.