ActivityPods can be thought of as a framework for building Fediverse apps on top of Solid. Solid itself is a standard developed by Sir Tim Berners-Lee, and once you get through the dry, somewhat brain-melting standards draft, what you get is something that strongly resembles a self-hostable Google Drive.
What makes this unique is that you can make apps that access resources (files) inside of this drive. Even more interesting is that Solid apps don’t rely on traditional databases in the way that regular Fediverse apps do: instead, the apps use files and metadata about those files, and basically parse information from a graph to act as a sort of rudimentary database instead.
What ActivityPods does is bridge the gap: it does all the heavy lifting to translate between Solid’s way of doing things (everything is a file) to ActivityPub’s way of doing things (everything is an activity performed on objects by actors, who have an inbox and outbox).
Why might someone want this? Well, the general idea is that this is a different approach to data ownership, where the data itself is portable as a file. Your Solid Pod acts as an identity provider that proves who you are, as well as a data store where you can decide who has access to read or modify your stuff.