API versioning is definitely a debatable subject, and I don't think that anyone at Stripe would claim that the current state of affairs is perfect by any means, but it's one that we think provides a good compromise between the stability of client integrations and our own ability to iterate on the API's design and make progress.
The classic problem with web APIs is that unless you have a good versioning scheme, once you've published them, you can never make a backward incompatible change (like removing a field) unless you're okay with breaking some people's integration. Especially when it comes to payments, people tend to have strong feelings about having their integrations broken, so we try to take as many precautions as possible to make sure that doesn't happen.
An approach to versioning that you'll see in many places is to do "major" API versioning where you do something like prefix your URLs with `/v1/` or send in a special `Accept` header. A problem with that approach though is that you'd need an incredibly good reason to ever build out a `/v2/` because if you ever bump that major version you're going to leave an incredible number of users behind on the original. Most people want to integrate one time and not have to worry about upgrading (ideally ever).
At Stripe, we've tried to build a compromise by introducing minor, date-based versions that include only a fairly constrained set of changes, but which we can bump more liberally. As others have mentioned here, your account gets locked into a version on its first request, and if never want to worry about API versioning at all, you can leave that version untouched essentially indefinitely.
If we realize that we made an API design mistake somewhere, we can fix it relatively easily and keep the API's design more cohesive for new users, while also leaving current users unaffected. It's also much easier to maintain for us because we only have to build a small compatibility module instead of having to maintain two (or more) completely divergent major API versions.
Anyway, I hope that helps explain some of the thinking behind this versioning scheme :)