how a flywheel can store energy in engines and other power producing devices

It does not store energy so to speak. The flywheel is a rotating mass that tends to stay rotating and help provide torque to the engine. The larger the flywheel the greater the torque of the rotating mass.

magnetism is stored energy

Through conservation of rotational momentum..

see newtons laws of physics.
http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion

If u really want to know this link will bring u through the math, http://geocities.com/budb3/arts/rpwr/flywhl.html however,
The flywheel is used to smooth out vibrations & continue motion.
How does it do this,
wow , l started typing this big long example & stopped & deleted it all, way to complicated. A simple example would be a heavy car & a light bike going down the road into the wind.
As the lite bike gets blown headon by the winds, the big gets slowed down & shudders, the bike rider then speeds back up when the wind isnt blowing again, & then the wind blows for a second or 2 again, the bike gets slowed doe=wn & then has to speed back up again. etc.
The heavy car has mass & momentium, so when the head on wind blows for a second or 2, the mass of the car is harder to overcome & it slows down little or none.
The same would apply to side winds, these small bursts blow the bike left & right, while not effecting speed they cause vibration on the bike , the car & its weight barely feel this if at all.
Now to add to that, theres always a trade off in energy. It takes more work to get & keep the heavy car moving, but its mass keeps it moving smoother.
A flywheel does the same, once moving, this big spinning weight over comes alot of the small variations that effect the speed of the engine or cause vibration & help keep it moving while minamizing vibrations.

a heavy flywheel will take more power to get it to rotate to a specified RPM, the more power it takes is stored as inertia in the flywheel. when you remove the power that was applied to get it rotating then the flywheel will release this inertia (energy) and continue to rotate.

if you have a light flywheel it will loose rpm much faster because it can not store as much inertia. and also took less power to get it to rotate to the same RPM as the heaver one.