I'm no startup expert, but I imagine part of it is that the company's owners (founder, VCs, etc) would like to reap some rewards of their successful endeavor and diversify their riches into other businesses. That means finding buyers, which is presumably most easily accomplished by going public. It's great to build a billion dollar business, but it might as well not exist until you sell some of it to someone.

I imagine that going public, simply by creating a lot of new shares, is also an effective way to raise additional capital, which could be used to grow the company more, similar to a private fundraising round but larger in scale.