Where can I read a proof of this?

Not sure whether this is needed anymore, but: The paper you want is John Steel's "The Derived Model Theorem." This paper gives a thorough and superb presentation of weak homogeneity and much more, including the result you are asking for. It is an unpublished note; the latest version is dated May 29, 2008, and can be downloaded from his page. However, you may want to take a look at "A proof of projective determinacy" by Martin and Steel, Journal of the American Mathematical Society, 2(1):71–125, 1989, although it may not look quite as what we are used to think of these things now. And probably you want to read first Martin's proof of determinacy of $\Pi^1_1$ from sharps, since the key ideas are there (this should be in Jech's or Kanamori's book). A modern, very quick and nice exposition of this and the key related ideas in proving determinacy at the projective level is in Itay Neeman's paper "Determinacy in $L({\mathbb R})$", in the Handbook of Set Theory. You can currently download the paper from Itay's page. 

