why does my bike chain fall off?

It could what the other guy said. But it could also be a worn crank bearing. Check for any unusual play in the cranks system.

This will cause a misalignment of the front and rear sprockets causing the chain to skip a tooth and peel itself off.

Also check for damaged or ramped sprocket teeth.
 
because your rear sprocket is not properly aligned with your drive sprocket, try playing with the rear wheel a little; i bet you will get it working properly
 
you need to adjust the tension on the chain, get a bike shop to remove a link of the chain to see if it grips better....the shop should be able to fix it
 
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