Why was there such a long gap between the parts of the "Before" Series?

Ethan Hawke discussed this with Collider:

He also revealed, “We had a version of [Before Sunset] that happened two years later. But once it took us five years and we can’t get the money to make that movie we have to rethink it. So elements or lines from that script stayed. When we first finished Before Sunset there was an idea that we would do. Julie and I wanted to do a short film the next year that we thought would be really fun to do like a five minute short and then put it in the vault so to speak and then fifteen years later make another film.”

The article also discussed the third film:

If you’re a fan (like me) of these two amazing movies and are hoping they’d do a third film, it sounds like they aren’t against it, but don’t expect to see it anytime soon. That’s because even though people constantly give Hawke ideas for the film, he says, “My thought is we will make one as soon as everybody has forgotten about it.”

They later clarified the exact reason they released the third film when they did, in an interview with French website Allocine. Quoting from The Guardian's rewrite of it:

Hawke reveals that he, Delpy and Linklater are negotiating a third outing for the transatlantic couple. "All of three of us have been having similar feelings that we're ready to revisit those characters," said Hawke. "There's nine years between the first two movies and, if we made the film next summer, it would be nine years again, so we really started thinking that would be a good thing to do. We're going to try to write it this year."

So to summarise, they wanted to release Before Sunset earlier, but struggled to get funding and that delayed the launch. When they did finally release the film, nine years had passed. They didn't want to release Before Midnight too early, and eventually settled on releasing it nine years after Before Sunset, to have the same timeline consistency between the first two films as the second and third films.