Newton is the man.
All you have to find is what does the air do when a wing passes through it. It accelerates the downwards and forward, so the air gives the wing an equal force upwards and backwards.
The more air there is, the more lift (either by increased density, wing surface or angle of attack)
the faster the air is, the more air is accelerated,
the faster the air is, the more accelerated it is
hence, lift is proportional to density, wing surface and angle of attack, and to squared speed. then you use the Lift Coefficient as the constant of proportionality, only it depends on the angle of attack, and that is the formula