In simple words, a message queue is where messages are stored until they're handled. Call it a buffer if you'd like. If you've ever used SDL, its events are put into a queue, which you pull the events from at a good time in your main loop. A message queue is required here because your program can't just stop in its tracks when ever SDL receives a new input and asks your program to handle it. Instead, you get the option to look at it later.

This is called asynchronous message handling. You fire off an event and it may be handled at what ever time the receiver thinks is a good time. Much networking works like this and things like Node.js make use of it to a great extent.

The opposite is synchronous message handling in which case something fires off the event and wont continue execution until the event is handled. No queue is required here, because the receiver will only handle one message because the sender wont be able to send more than one message at the time.

Apart from being essential for networking and multi-threading, a message queue can also be used if you want to send events that wont be handled until the next game turn or "tick". Of course, a message queue can be polled at less frequent intervals.. I don't have much say in that. But the most normal use would be for it to handle events each iteration of the game loop.