Flaps are not required to take off. Flaps change the amount of lift (and drag) the wing is producing. This allows for lower takeoff speeds and consequently lowers the distance required to take off. You're right about Bernoulli's principle, but the lowering of flaps is probably closer to Newton's third law, which is also a valid explanation of lift. Air flows over the top of the airfoil and is accelerated downward, and the reactionary force is upward on the wing. Neither Bernoulli's principle nor Newton's third law is sufficient to fully explain lift on its own, there are elements of both.
But that's a bit more detail than I've ever been expected to know on the subject. Knowing the effect of flaps on approach/stall speeds and descent profile, sure. Pressure differentials and reactionary forces? Not so much. For more practical knowledge, check out Ch. 4 of the Pilot's Handbook of Aeronautical Knowledge (the whole thing is free to download online):
http://www.faa.gov/regulations_policies/handbooks_manuals/aviation/pilot_handbook/media/PHAK - Chapter 04.pdf