My initial attraction to Stripe was because of a reason mentioned in the post: not wanting to redirect or otherwise interrupt the normal order process of my site with another company's (branded) checkout form just to handle credit card payments. The other reason for initial attraction was of course their elegant API, which was very refreshing to see after having dealt with Intuit's QBMS API (shudder) in the past.
Stripe really sealed the deal though when I tweeted them to ask my first question about signup and one of the co-founders reached out to me. Since then, I've been in contact to share my feedback or ask other questions, and the quality and timing of their responses is one of the best I've ever seen.
When a customer of ours had an issue placing his order, I wrote Stripe to see if they could offer any insight as to what errors they might be seeing on their end. While the error was unrelated to Stripe itself, they were so awesome that they actually took the initiative to browse through our checkout page and point out the spot that they thought to be the issue.
A company that stands beside you when debugging your checkout page, maintaining swift response times on an issue that isn't even theirs? That's customer support.
I probably sound like I'm gushing right now, but I just have mad respect for this company. The product isn't the only thing that's top notch, the people behind it rock too.