The way it was explained to me is that every Lemmy instance is basically a full on “reddit” in that it’s a link aggregator, supports user made communities (ie: subreddits), commenting, etc. You can run Lemmy in private mode and this is exactly how it functions!
On the side of what “federation” is, it’s that all the instances can (theoretically) communicate with each other and share posts and content amongst themselves. So let’s say you make a post on lemmy.world, because my instances “federates” with lemmy.world I am able to see your post and comment on it from my instance. Lemmy.world and my instance periodically update each other with posts our respective users make. Your post lives on Lemmy.world, my comment replying it to lives on mine, and when I post my comment Lemmy.world receives a notice that I’ve done so, which then creates a notice for you that I’ve made the comment blah blah.
The benefit to federation mainly is that it gives a lot of control to users on how the platform functions. Firstly it doesn’t congregate the entire userbase to a single company and/or site. No single instance should remotely be as large as reddit. But because they communicate together, you can approve/deny what instances (as an instance admin) you’re “federating” with. Don’t like the users and moderation policy of another instance? You can “de-federate” with them and block their content from showing up on your instance.
Of course they’re expensive think of how much labor it takes to put them in the damn shells!