a matrix M is said to be positive definite if for any vector v, v^{T}Mv ≥ 0

note that the inner (dot) product of two vectors u,v can be written as the matrix product u^{T}v.

now v^{T}(A^{T}A)v = (Av)^{T}(Av) = (Av)•(Av).

(for any two matrices, it is a property of the transpose that (AB)^{T}= B^{T}A^{T}).

but for any vector u, u•u ≥ 0 (this is a property of inner products, called positive-definiteness as well),

hence v^{T}(A^{T}A)v = (Av)•(Av) ≥ 0.

(by the way, the proof you show is one that shows AA^{T}is positive definite).