Manuals actually have a flywheel - flexplates are specific to automatic transmissions. Regardless - either setup has an engine, which needs centrifical mass to keep turning.
Clutches on standard tramissions don't have flexplates, they have a flywheel. A big spinning disc with a machined surface. It's job is to keep the engine turning with its inertia between ignition of the pistons, as well as make the clutch easier to operate (imagine an engine with no power right when you're getting started off the line and engaging the clutch - it'd be impossible to get going unless you roll down a hill first). It's other job is to provide a surface for the friction plate to sandwich between. The flywheel shares this job with the pressure plate, which bolts to the flywheel and spins with it, and pushes the friction plate into the flywheel as well as itself, engaging the plate to the engine when under pressure (foot off the clutch). This in turn allows the transmission to get power as it's splined into the friction plate.
An automatic obviously doesn't have a clutch - it's permanently engaged via a viscous coupling (or mechanical if you have a lock up converter and are driving on the highway with it engaged). This system relies on pressures in the flow of the transmission oil to spin two turbines essentially. When you spin the engine, the torque converter housing (turbine 1) spins, and this imposes pressure through the transmission oil on the turbine splined to the automatic tranmission (turbine 2). Because it's not a direct connection - you can stop turbine 2 without imposing too much of a load of turbine 1 at slow speeds. At higher speeds the pressure becomes so great that slippage tends to get eliminate (look into stall speed of torque converters)
Think of it as mixing dough in a bowl. You spin the mixer and hold onto the bowl. The dough whirls around the mixer's blades. If you let go of the bowl and keep the mixer going - the bowl will begin to spin with the mixer through imposed pressure because of the thick viscosity of the dough.
Regardless of how it works - the engine still needs something to keep it spinning between ignition - even with an auto. So that's what the flex plate does. It's the flywheel... for an automatic transmission. It doesn't need a machined surface like a clutch flywheel does, but it does need mass.
Fly wheels and flexplates look similar - a big round disc with teeth on the side for the starter to engage to, but the flexplate usually has less material since it doesn't need to provide a surface for a friction disc to bite into, spin, and wear away at.
Note the smooth surface where the friction plate engages, and the threaded holes around the perimeter for the pressure plate to bolt to. Some flywheels also have a removable mating surface for easy maintenance (buy a new surface insert instead of remachining)
note the stamped metal construction, with bolt holes for bolts to go through and mate the flexplate to a torque converter.