The fallacy is thinking that this was the puzzle. This is just one piece in a puzzle that's so big no one person can see the whole thing, and that we're using as a blueprint for large parts of modern civilization while still figuring it out. We don't even know whether we've completed most of the puzzle, or whether the puzzle is actually vastly larger than what we've already put together.

If you can't see why this is useful, then great: that's literally what it means to be a layman. But that also means that you don't know enough about the subject field for the real answer (the one that explains what other maths this affects) to satisfy your question because you won't know why _those_ things would matter.

So you're going to get the layman's answer: "because this shows that any work already performed in the assumption of its truth is now known to be valid", which is critically important because a lot of what maths is used for in real life is based on assumptions about the properties of numbers. Now others can stop wasting their time trying to determine whether this aspect of number theory can be relied upon, and under which circumstances, and trying to find all expressions of it in both software and hardware in order to determine whether those might have problems on a case-by-case basis. And at the same time, this result allows others still to move forward with work that is known to rely (in part!) on this property either being true or false, for which it was previously uncertain whether it would even be worth the effort because it was unknown whether the results would turn out to be useless because no one knew whether the basis it relied on was flawed.

As for what useful development comes out of things like this:

points at the computer you're using to comment on HN threads, and every single application on it that you've ever used

That.