Then what are they running their website on? This is a perfect example of the infectious nature of the word cloud, people say the word cloud as if there's no server still running it!

Oh noes, I have to run php code stripe has done all the work on and provided to me! Scary!

There's a big difference between running a server that does basically glorified inventory tracking, and running a server that hosts (even just a wrapper around) payment infrastructure.

If my inventory service breaks I might have some pissed-off customers or have to re-ship some items, the odds of getting in bigger trouble than that are low. If my payment infrastructure goes down or gets compromised, the odds of lawsuits or federal/financial-org penalties are way higher.

So sure, you still probably have to have a "server" somewhere (you could probably make a fully static site for a merchant using third-party payment links, javascript, and mailto links or something--think "email as database"--but I suspect this would be neither featureful, secure, nor reliable), but the difference in what you need to maintain/serve is huge.

Working with Stripe and PayPal are essentially the same if you use best practices. Stripe can do the payment page and redirect if needed and can also capture paying info from a static page that is submitted via JS in the background. In the end you'd get a simple token that's just a reference to the payment details in Stripe (just like PP). If you integrate with the Stripe payment page then they will even remember you across sites (again, just like PP) and you won't have to input payment details.

If you use Stripe, you still need a server to make secure API calls to charge the card. The card is not charged until you make an API call with the token.

Couldn't you run a more static based site if you don't have to run some PHP code though?

I was just using php as an example. See stripes documentation for more info:

https://stripe.com/docs/stripe-js

https://stripe.com/docs/charges