The neat row of teeth scribed around the edge of the flywheel mesh with a small gear in the starter motor, which typically is bolted to the front of the engine just behind the flywheel. When you turn the ignition key to the Start position, the starter gear shoots forward from the starter motor housing and engages the flywheel. The starter motor spins the starter gear, which turns the flywheel, which turns the crankshaft, which forces the pistons up and down in the cylinders. When a fuel-air mixture enters the cylinders and is ignited by the spark plugs, the engine fires and runs.
Once the engine starts, the flywheel's job is just beginning, however. Unlike a turbine engine that has continuous ignition of the fuel-air mixture, each cylinder in an aircraft piston engine fires in sequence, like a machine gun. The mass of the flywheel acts to smooth out the engine's relatively rough power pulses.
There's more. On most light aircraft, if you look at the back side of the flywheel you'll see a belt that rides in a large groove. Follow the belt and you'll see that it wraps around a small pulley on the front of the alternator (or possibly generator if it is an older aircraft). In other words, the flywheel provides the motive power to drive the alternator.
So the next time you're preflighting the airplane, reach in past the spinner and give that flywheel a tap of appreciation for its versatility and impressive workload.