Think of Fediverse as akin to the email system, where there are thousands if not millions of different servers out there hosting email and exchanging them to each other.
Now consider Mastodon as an email client which is hosted on a web site, like Gmail or Yahoo! Mail or Hotmail. That web site gives you an interface with which to interact with the network of email servers, but people using other servers with other interfaces can still see messages from each other; someone using Yahoo! Mail can send and receive messages with someone using Gmail.
The difference is that Mastodon is open source, so that anyone can download the code and run their own Mastodon instance (whereas nobody can download and run their own Gmail instance), but that instance can still exchange messages with instances running any of the several other Fediverse interfaces there are out there, such as Pleroma, Misskey, Soapbox, etc. (Unless an instance at either end has "defederated" the other; that is, the administrators have configured it to not exchange messages with the other server.)
E-mail, although it has different clients, is still fundamentally the same thing — you send a message with a subject, body and attachments. However, it seems that social networks built on ActivityPub are fundamentally different — Mastodon is an imitation of Twitter, with short low-effort posts, Friendica is an imitation of Facebook, then there are ways to link blogs… So what I don't understand is, how does this all work together when the formats are completely different?
I don't have an answer for that since I'm not an expert with the protocol and I've really only used the microblogging/"Twitter-like" aspect of it. But at least in terms of that, the front ends are compatible with each other when it comes to sending and receiving messages.
Now consider Mastodon as an email client which is hosted on a web site, like Gmail or Yahoo! Mail or Hotmail. That web site gives you an interface with which to interact with the network of email servers, but people using other servers with other interfaces can still see messages from each other; someone using Yahoo! Mail can send and receive messages with someone using Gmail.
The difference is that Mastodon is open source, so that anyone can download the code and run their own Mastodon instance (whereas nobody can download and run their own Gmail instance), but that instance can still exchange messages with instances running any of the several other Fediverse interfaces there are out there, such as Pleroma, Misskey, Soapbox, etc. (Unless an instance at either end has "defederated" the other; that is, the administrators have configured it to not exchange messages with the other server.)