Rach has it down pretty nicely: you do something to one or more groups, and have another group that you do nothing to (or, in medicine, appear to do something to but don't really, with it either a being a sham [e.g. a placebo or a known to be irrelevant chiropractic manipulation] or the usual and customary treatment).
Now the reason is mathematical. In the real world, there are lots of variables that might affect the outcome of your experiment, but that you either cannot control, or cannot control for, or are totally unaware of. (Let us not kid ourselves, we don't know it all--not by a wide margin.) If you have a control group and an experimental group or groups, you can use the control group as an approximation of what would have "happened anyway" in your experiment, letting you tease out (roughly) the effect of whatever you did from what would have "happened anyway."
The explanation is sloppy, but I think it makes reasonable sense. Well, it does to me, anyhow
BA-Math with minor in Prob and Stat