Musicians need to be able to play in time. It is a skill many struggle with and is essential for a tight performance. To do this you must be able to hear the music as it is being played.
You may wonder why the main PA system isn't enough. In some circumstances it may well be. An example of this is a tiny stage in a tiny venue. But what about larger venues?
The thing is, sound takes time to travel. If the PA system is in front of the band, pointing at the audience, the sound has to travel all the way to the back wall before bouncing off. It then has to travel all the way back, past the speakers before it reaches the band.
While sound does travel fast, it doesn't take much time delay at all for it to be out.
Consider the delay is 0.2 seconds. Not really long but it is enough. If I tap on my spacebar as fast as I can go, I can hit it once every 0.13 seconds and there is a distinct audible difference between each tap.
This is why you want monitors. They are right close to the performers and so there is a negligible time delay between input and hearing the sound.
You can see here, the monitors right in front of the band,
angled up so they can hear. (of course, this is Gorgoroth so
it may as well be pink noise fed into those monitors!)
Another reason for having monitors is the requirements of the band members. By having a monitoring system, each band member can listen to what they need, at the volume they want. For example, the bassist is going to want to be really tight with the drums and so he will most likely have the drums turned up in his monitor. The guitarist may want bass, drums and, vocals but the recorder may not be necessary to hear, and so, the former would be louder while the latter can be turned down.
Monitors can be controlled via aux sends on the live mixer but in larger scale concerts, a dedicated monitor engineer may be used whose sole job is to make the levels right for each band member.
All in all, it is a very important component to your live setup and, even just one or two monitors facing the band may make a big difference to the show.