Previously you had a few choices, go with:

1. PayPal, who wanted to own the customer relationship

2. A payment processor who all had awful APIs and you had to do lots of setup around having a merchant account. Authorize.net I think was one of these. Recurring payments were often a nightmare to get to work.

3. An intermediary processor, who didn't need you to have a merchant account, nochex, Worldpay, sagepay, etc. who again generally all had awful APIs and hard to embed widgets and complicated setup processes. Again, recurring payments a nightmare.

Stripe solved all these pain points, and the API was great. It wasn't one pain point they solved, it was many.

You controlled the customer relationship, it was easy to embed the stripe widget, it had a great API, you didn't need a merchant account and the setup was quick and easy. Recurring payments were (fairly) easy to set up. Oh, and great documentation! That was another big thing.

And then they brought out a little widget you could attach to your phone/tablet to allow easy in-person card payments!

It probably seems a bit incomprehensible now, but this was a time when everyone did everything slightly awkwardly. Integrating with any third party was a big job. Documentation was terrible. Formats were all over the place. SOAP, XML, JSON was only just emerging as the defacto standard. For example, the APIs. Great APIs were rare, as far as I remember twilio and I think MailChimp really blazed the way there. Companies like Google were putting out absolute dogshit, complicated APIs, often strict REST with weird header requirements, or strange signing with private keys. If you've never dealt with it, strict REST really, really sucks.