The roach allows the sail to form a curved surface, creating an airfoil, and threrefor generating "lift", just like the wing of an airplane, or the rotor blades of a helicopter. The physics involve Bernouli's Principle, best illustrated with... an illustration
The Luff and Loach of the sail are equivalent to the leading and trailing edges of an airlplane wing, and the straight lind distance between those is the "chord". (also, the aeronautic equivalent of "roach" is "camber")
The air flowing past the sail has to flow faster on the convex side of the airfoil, which results in lower pressure, which is called "lift". The"lift" is more or less perpendicular to the chord, so of course in the case of the sail as airfoil, the "lift" pulls the sail and boat forward, rather than up.
If you think about sailing on any point higher than a beam reach, you can see that it is the lift generated by airflow over the curved shape of the sail, and not increased sail area that is propelling the boat forward. It could be argued that increased sail area alone is only really advantagous when running downwind, wing and wing.
Does that help?