a matrix M is said to be positive definite if for any vector v, vTMv ≥ 0
note that the inner (dot) product of two vectors u,v can be written as the matrix product uTv.
now vT(ATA)v = (Av)T(Av) = (Av)•(Av).
(for any two matrices, it is a property of the transpose that (AB)T = BTAT).
but for any vector u, u•u ≥ 0 (this is a property of inner products, called positive-definiteness as well),
hence vT(ATA)v = (Av)•(Av) ≥ 0.
(by the way, the proof you show is one that shows AAT is positive definite).