I understand how that works qualitatively, but I don't understand where the number 99.9999% comes from, nor how the number 25000 feeds into it. And surely only a tiny fraction of mathematics papers on the arXiv deal with conjectures of this sort, so why cite the number of papers there in particular?

I'll be (rather bogusly) quantitative for a moment. Pretend that every single one of those 25k papers on the arXiv makes an argument similar to Euler's, and that if any one of them were wrong then you'd certainly have heard about it. How improbable would an error have to be, to make it unsurprising (say, p=1/2) that you haven't heard of one? Answer: the probability of one paper being correct would have to be at least about 99.997%.

Now, to be sure, there's more out there than the arXiv. But, equally, hardly any papers deal with arguments like Euler's, and many papers go unscrutinized and could be wrong without anyone noticing, and surely many are obscure enough that even if they were noticed the news might not spread.

Maybe I'm being too pernickety. But it seems to me that one oughtn't to say "In the context of the fact that ~25,000 math papers were posted on ArXiv in 2012 it may be reasonable to conclude that the appropriate confidence level would have been 99.9999+%" when one means "There are lots of mathematics papers published and I haven't heard of another case of something like this being wrong, so probably most such cases are right".