I would like to add to this another perspective.
Stateless authentication without hitting the DB on each request
Let's suppose you want to create a stateless (no session) security mechanism that can do authentication of millions of users, without having to make a database call to do the authentication. With all the traffic your app is getting, saving a DB call on each request is worth a lot! And it needs to be stateless so it can be easily clustered and scaled up to hundreds or even thousands of servers.
With old-fashioned sessions, the user logs in, at which point we read their user info from the database. To avoid having to read it again and again we store it in a session (usually in memory or some clustered cache). We send the session ID to the client in a cookie, which is attached to all subsequent requests. On subsequent requests, we use the session ID to lookup the session, that in turn contains the user info.
Put the user info directly in the access token
But we don't want sessions. So instead of storing the user info in the session, let's just put it in an access token. We sign the token so no one can tamper with it and presto. We can authenticate requests without a session and without having to look up the user info from the DB for each request.
No session ... no way to ban users?
But not having a session has a big downside. What if this user is banned for example? In the old scenario we just remove his session. He then has to log in again, which he won't be able to do. Ban completed. But in the new scenario there is no session. So how can we ban him? We would have to ask him (very politely) to remove his access token. Check each incoming request against a ban list? Yes, would work, but now we again have to make that DB call we don't want.
Compromise with short-lived tokens
If we think it's acceptable that a user might still be able to use his account for, say, 10 minutes after being banned, we can create a situation that is a compromise between checking the DB every request and only on login. And that's where refresh tokens come in. They allow us to use a stateless mechanism with short-lived access tokens. We can't revoke these tokens as no database check is done for them. We only check their expiry date against the current time. But once they expire, the user will need to provide the refresh token to get a new access token. At this point we do check the DB and see that the user has been banned. So we deny the request for an access token and the ban takes effect.