Real flaps increase the coefficient of lift greatly and the AOA during take-off, making the same wing's area lift more at the same velocity or lift the same (weight of the plane) at slower velocity.

The roll distance to reach certain velocity is a direct function of the thrust minus drag.

Flaps increase drag, increasing the distance at which the old take-off velocity (no flaps) is reached.

However, as the take-off velocity with deployed flaps is lower, the take-off roll distance with moderately deflected flaps is normally shorter than without flaps (for the same thrust force).

A caveat: the flight velocity with deployed flaps is lower than the flight velocity with flaps off; hence, velocity must be increased while flaps are turned off, or the wings will be unable to sustain flight.

This article explains it better:

http://www.airdock.cz/doc/teo/flaps.pdf