I recently learned about CORS and got the impression that its purpose is to prevent XSS. With CORS, the browser blocks requests to different domains, unless particular headers are in place.

But if a person with malicious intent injects some JavaScript into a page to steal users' cookies and send them to a URL he controls, all he has to do is add the following header on the server side to make the request work anyway:

Access-Control-Allow-Origin: *


So how does CORS prevent XSS? Or did I misunderstand the purpose of CORS, and it simply has nothing to do with XSS per se?

TL;DR: How does CORS prevent XSS? It does not. It is not meant to do so.

CORS is intended to allow resource hosts (any service that makes its data available via HTTP) to restrict which websites may access that data. 

Example: You are hosting a website that shows traffic data and you are using AJAX requests on your website. If SOP and CORS were not there, any other website could show your traffic data by simply AJAXing to your endpoints; anyone could easily "steal" your data and thus your users and your money.

In some cases that sharing of data (Cross Origin Resource Sharing) is intended, e.g. when displaying likes and stuff from the Facebook API on your webpage.
Simply removing SOP to accomplish that is a bad idea because of the reasons explained in the above paragraph. So CORS was introduced.

CORS is unrelated to XSS because any attacker who can place an evil piece of JavaScript into a website can also set up a server that sends correct CORS headers. CORS cannot prevent malicious JavaScript from sending session ids and permlogin cookies back to the attacker.
