> it's difficult for me to see exactly where it would make my life easier or better
As a shopper it won't. Stripe is not targeted at shoppers but at website developers who want to implement credit card payments in a sane and easy way.
As a user, you should not care whether the website is using Stripe or not (although their UX is usually nicer by default and their development guidelines help the implementation be more secure). The added value to you is that it is easy for the developer to implement credit card payments the right/secure way, therefore giving you more payment options.
As a developer, I welcome this open armed. I've come into contact with several payment gateways (GMO/Econtext/F-regi) while working in Japan, and all of their API's have been an atrocity, with lackluster documentation.
As a shopper it won't. Stripe is not targeted at shoppers but at website developers who want to implement credit card payments in a sane and easy way.
As a user, you should not care whether the website is using Stripe or not (although their UX is usually nicer by default and their development guidelines help the implementation be more secure). The added value to you is that it is easy for the developer to implement credit card payments the right/secure way, therefore giving you more payment options.