Stripe was born to make payment easier for developers. Other payment services, such as PayPal, didn't focus that much on developers.
Patrick Collison knew Paul Graham from Y Combinator. Collison's previous project, Auctomatic (an online auction management system), was built thanks to YC.
Before Paul and his brother, John, launched Stripe's prototype, it took them 6 months to iterate a lot and ask their friends for feedback. Then, they had a prototype ready in 2 weeks. 280 North was their first customer. It was one of the YC companies.
Graham trusted Patrick with Stripe because Auctomatic was profitable as well. Around $30,000 was enough for Collison to get more traction. /Dev/Payments was the primary name, but it was changed to a much simpler Stripe pretty fast.
Stripe grew mainly through word of mouth. In Patrick Collison's words: "That was surprising to us because it’s a payment system not a social network so it’s not something you’d think would have any virality whatsoever. But it became clear that everything else was so bad and so painful to work with that people actually were selling this to their friends."
Other than that, Paul Graham's activity on his social media channels (mostly Twitter). He put great trust in that project. Stripe was also talked about on the YC blog.
[source]
[source]
[source]