I know Major Macmahon conjectured the formula $$ \prod_{m=1}^\infty \frac{1}{(1q^m)^m}=1 + \sum_{n=1}^\infty PL(n)q^n$$ but who was the first to prove it?

The answer is MacMahon himself, who proved this in his book Combinatory Analysis as a corollary of a more general theorem about plane partitions. See Sections IX and X. There is some additional historical information in the Notes to Chapter 7 of Richard Stanley's book Enumerative Combinatorics, volume 2. 

