Comment by ChadNauseam

The other commenters point at the explanation but don't explain it rigorously IMO. Here's how I'd say it.

60% is a nice, round percentage. In an honest election, this is just as likely to be reported as any nearby percentage, like 59.7% or 60.3%. As you mention, any particular percentage is equally (and extremely) unlikely. SUppose this you estimate the chance of this occurring, given an honest election, is 1/1000.

60% however is a much more likely outcome if the election results were faked sloppily. A sloppy fake is reasonably likely to say "Well, why not just say we won 60%". Suppose you estimate the chance of this occurring, given a sloppily faked election, are 1/100.

Bayes' theorem tells us that we can use this information to "update our beliefs" in favor of the election being faked sloppily and away from the election being honest. Say we previously (before seeing this evidence) thought the honest:faked odds were 5:1. That is, we felt it was 5 times more likely that it was honest than that it was sloppily faked. We can then multiply the "honest" by 1/1000 (chance of seeing this if it was honest), and the "faked" by 1/100 (chance of seeing this if it was faked), to get new odds of (5 * 1/1000):(1 * 1/100), which simplifies to 1:2. So in light of the new evidence, and assuming these numbers that I made up, it seems twice as likely that the election was faked.

This exact analysis of course relies on numbers I made up, but the critical thing to see here is that as long as we're more likely to see this result given the election being faked than given it being honest, it is evidence of it being faked.