The only reason CSRF is even possible is because the browser sends (or, well, used to send) cookies for a particular request even if that request initiated from a different site. If the browser never did that (and most people would argue that's a design flaw from the get go) CSRF attacks wouldn't even be possible. The SameSite attribute makes it so that cookies will only be sent if the request that originated them is the same origin as the origin that originally wrote the cookie.

I think I understand now, the Cookie just is not present in the POST if a user clicked on, for example, a maliciously crafted post from a different origin?