Basic calculus (if there is such a thing !!!)
Let me try for a simpler version of Matt's answer, and a more correct version of DD's answer.
The lift depends on the kinetic energy of the free stream
From physics we have:
momentum = mass x velocity, and
kinetic energy = the derivative of momentum
From any calculus 101 textbook:
If momentum = mv, then
kinetic energy = 1/2 mv^2
Now substitute rho for m, and you get the answer as it applies to free stream aerodynamics.
Does that help ? Dunno !! It's still simplified (a lot) but it's more accurate.