The logic behind "better safe than sorry"

My understanding of the phrase is that it is used in a situation which has some significant probability of becoming adverse without proper attention or prevention. It's an argument in favor of taking preventative action. The saying indicates that one would be wise to take appropriate action to make sure that undesirable state doesn't come about.

better [to take the time/effort do this and ensure you are] safe than [to not do this and possibly become] sorry.

It's typically a suggestion, and more risk-prone folks may choose not to heed it and still be okay. For example:

I'm going to ride my motorcycle without a helmet because I enjoy it much more that way, and there's only a small likelihood that I'll get into an accident. <--- This person incurs a higher risk of death, but enjoys the ride more

I would rather wear my helmet when I ride and have a greater chance of surviving if I do get into an accident. <--- This person forgoes the better feeling for extra protection against a deadly outcome

You could apply this to many situations: safe sex, pop quizzes, etc

This is like "it's better to buy insurance and not need it (than it is to not have insurance and need it)."

In this phrase, being safe requires effort to be in that condition, but the effort is small compared to what loss might occur if that effort weren't made.

Examples are:

It's better to check behind your car every time you back out of your driveway, even if you don't have children or pets.

It's better to always check if the gun is loaded, even if you don't have bullets.

This axiom is not a comparison between safe and sorry. It is a reminder, born out of bitter experience, that adverse situations will occur, and being prepared for when they do is better than the alternative.

Safety always takes time and effort that many see as a waste. If an adverse action occurs infrequently, it is all-too-human to assume that just because it hasn't happened recently, it will not happen.

As you've explained in your question, "better safe than sorry" is not a comparison between two things someone has to choose between, because if it was it would be incredibly one-sided. Everyone agrees that it is better to be safe than sorry. Instead, it's an explanation, motive, or argument for doing something -- because in the future, we want to be safe rather than regretting that we didn't do something else or take some extra step.

In use, it's a short way of saying the much less convincing "You/I might end up regretting it if you/I don't do this." This phrase is mostly valid when the extra step has no risks, but the person you are telling the phrase to believes that the extra step is unnecessary, paranoid, or just overkill.

Using this phrase might be analogous to:

Me: Why are you eating an apple?

You: Humans get hungry.

I already knew that humans get hungry, but you've still managed to communicate some new information for me (that you were hungry) by responding that way.

It struck me that the phrase "better safe than sorry" is [...] so logical and obvious [...]

You're quite right.

Better safe than sorry

The idiom's meaning can be explained by an expansion on the words themselves. I would more didactically put it:

It is better to be safe by taking a more prudent course of action than to take the riskier course of action and be sorry for it when facing its consequences.

The statement is not as obvious at it might seem and it is trying to combat an inbuilt psychological resistance to facing harsh or unpleasant realities.

I think that there are an astonishing amount of accidents with high human cost that are caused by the brain's in-built forgetfulness or blindness to risk. In order to keep us happy and sane, our brains forget or ignore a lot of things that if we stop and think about them would make us unhappy. You only have to see the news to see how many accidents are caused that could easily have been prevented, but occur because one person thought that 'it wouldn't happen to them'.

Therefore the phrase is designed to instruct us to combat that in-built resistance with an always present compensation to be 'safe' and it explains that we should because the consequences (although probably rare) might make us very sorry.

There are a lot of people out there who have injured or killed other people through negligence because they didn't exercise appropriate caution. They live their lives always knowing that they should have been more cautious and this phrase is simple and could save a lot of sorrow if remembered.

Obviously this phrase applies to many less important issues, but the core message is 'always remember that something can go wrong that you would regret'. Whether remembering it saves a life or just makes you stop to fill up the car with fuel earlier, it helps us remember that we need to show extra caution because our brains are wired to be less cautious than we should be.

The logic is that most people have loss-averse utility functions, as demonstrated by Tversky and Kahneman. This is because there is an asymmetric emotional reinforcement incurred when comparing a loss of magnitude X with a gain of magnitude X. To maximise expected utility across all future states of the world, given by the objective function {argmax E[U(w)]}, agents will therefore seek a solution E[w] < 0; i.e., being 'safe' instead of 'sorry'. To justify otherwise, human agents need a mass function of future reinforcement whose integral from infinity to zero is more than half of the total area under the density (note that I refer to monetary reinforcement instead of the emotional reinforcement that has already been pushed through function U(.)).

It is about reminding someone to consider something that they might otherwise overlook: to be "safe" by choosing to take some extra precaution is a choice you can make. If you don't, the possibility of being "sorry" because something that you didn't choose "happens to you", and you cannot undo that. I think it is about trying to have some control over what you can, in a world where we cannot control everything. This is sensible. The two possibilities are not equal, because you either take action, or do not. "Prevention is better than cure" amounts to the same thing: prevention is an action, doing nothing is not.