What normally happens is that you become too big and that's when VCs step out of the way, you IPO and everyone takes their gains. But in the case of stripe you have $100Bn company that the VCs will still happily plough money into. So there's no reason to IPO. You can get all the liquidity you need privately. On top of that, Stripe doesn't really need capital, they aren't some uber buying market share with loss making. They are real SAAS, they can scale and see a drop in their earnings:cost ratio.
As for why stay private, what John C has said publicly is that they still see a lot of growth to go, and that not having to provide detailed financian statements to the general public, which would include competitors, is a competitive advantage. It seems believable to me.
The only downside for the company seems to be that employees would be able to sell, and that seems rather greedy for the company to not work in its employee's best interest.