Some user wants to fetch a content that's on different server. Let's call the user's instance "Home instance" and the other one "Target instance".

Let's assume the Home instance and Target instance are not federated yet.

So, the user triggers the discovery of the Target instance on the Home instance. It varies by platform, but on both Lemmy and Mastodon it's done by searching for a URL (for example https://lemmy.world/c/protonprivacy). Because the instances don't know about each other, they now have to federate.

Federation basically means, that the Home instance subscribes to all changes by some part (or all) of the Target instance, in this case to a single community (to save space, traffic etc. they don't usually subscribe to stuff that no user is subscribed to).

Now, whenever you add a content (post, comment, toot on Mastodon), your instance checks the list of subscribers and notifies every single one about the change.

So if there's a new post here, the Target instance now notifies the Home instance that there's a new post, the Home instance downloads the data about the new post and makes a local copy. Let's assume the user makes a comment on that new post, in that case the Home instance notifies the Target instance that there's a new comment and the Target instance makes a copy.

That's basically how Activity Pub works.