The US was able to get cooperation because it made credible threats.
Specifically a 30% withholding for non-complying institutions. The US financial system touches a large part of the world, so for many institutions it could enforce meaning loses, and certainly make them non-competitive in the huge US market.
This is an issue on a national level too. If the withholdings went through on a large scale it could destroy economies. The same logic as other sanctions applies, the US having a bigger economy could be expected to better survive the disaster than most other places, and we are bold enough to try it.
So far as I can tell no one explicitly says they capitulated out of fear of these withholdings, but it is featured in some government explanations of why it matters. (Canada and Honk Kong)
Also it is not quite one sided, the US agreed to reciprocity in many cases, which might work out to a tax win especially for places with a wealth tax or with a large population in the US.