To be able to handle as many users as Twitter do, they have to make use of a mechanic called load balancing. That means Twitter does not run on a single server, but on many servers, and every time a user visits Twitter, it sends the user to its least busy server.
This load balancing setup has to be managed according to the amount of users Twitter expects. When Twitter expects many users, they typically start up more servers, and when they expect fewer users, they shut some of them down. This is because server time costs money - they would typically pay for each running server per hour.
Now, sometimes, they'll get these estimates wrong, which means more users are trying to visit Twitter than there are running servers for. That is when you'll get the over capacity message, which will last for as long as it takes them to start up new servers.