As I understand it, it goes something like this:

1. You are on lemmy.cafe and ask to follow [email protected].
2. lemmy.cafe asks to follow [email protected]
3. kbin.org makes a note to update lemmy.cafe every time someone posts to the cats magazine (but not previous posts).
4. Someone else on lemmy.cafe looks at the 'global' feed.
5. lemmy.cafe shows the [email protected] posts, along with all the other remote posts that were requested, and all the local posts.
6. You upvote a cute cat post.
7. kbin.org interprets this as a 'like' (or whatever)

The language these services share is called 'ActivityPub'.