What that boils down to is (sorry, Ms Lauper) DJs just wanna have fun. Do they *need* these things to make their mixes? Probably not as most of it gets developed over headphones or small studio monitors/sub... but when it's party time there is a show to do and fun to be had. I suppose at some point the DJ monitor rig is part of the stage set/props as well and becomes part of the *experience* for the audience, too.
Every DJ that I have ever worked with over my 17 year career uses monitors to sync the sounds in their headphones to what is coming out of the main PA. This process is called "beat-matching" and is fundamental to their performances. Whether using turntables or cd players, the pitch is adjusted so that all sources are at the same tempo and then mixed back and forth. Sources might need to remain synced live in the mix for minutes at a time.
Strong monitoring is required because any sound coming from the main PA is inherently delayed. Latency is a valid issue for musicians to stay in time with each other, how much delay is there when the sound reflects off the back of the room?
Yes, it is a volume game that the dj actually always loses. Anyone playing out longer than 6 yeas has generally lost some hearing, not unlike rock musicians. As loud as their monitors need to be over the din of the main PA which needs to be louder than the ambient noise, their headphones still need to be a lil louder to their ears.
There have been some advances with automatic software synchronization, but they are not perfect and most performers serious enough to have riders and play stages where a truck of gear is hired are still adjusting tempos manually. Big monitors are not for the audience, they care more about the lights.