No, it means if you run Lemmy as a service and make modifications to it, you have to release your modifications back with the same license. Otherwise you couldn't use a browser that's not AGPL and read pages running on top of an AGPL server.
What AGPL is really good at is how nobody can take Lemmy, run a proprietary service and add incompatible features without giving them back to the community. So nobody can fork Lemmy, create a new VC-backed Reddit clone and start making incompatible changes to the source without the main project getting the source code.