It was early Monday morning, you looked at the clock, and despite the fact you didn't have work, you got out of bed, and rolled into the gym. Why? Because you promised someone you would. Having a group of people who are expecting to see you (either a gym buddy, or even just the coach) can keep you committed and in the gym.
Even though we are all working toward our individual fitness goals, working in a class environment creates a natural team dynamic. The camaraderie that comes with this can be integral to your workout.
Once you walk through the doors, everyone in the class is there to suffer along side you. When you don't feel like you can do another rep, you might look over at Mr. X, who is breathing just as hard as you, but pushing through it, so you hunker down and keep moving. Or on that particularly sluggish day when you feel like you'll never get through it and everyone else seems to fly through the WOD, the rest of the class cheers you on to the finish.
Then there is the natural competitive spirit that creeps out and pushes you to chase or beat someone else in class, even if its just by one second. Friendly competition is healthy and encourages you to give 110%, and keeps you from letting your fatigue dictate your effort.