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Where can I read a proof of this?

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    $\begingroup$ Without the assumption that there is a Woodin cardinal, this can fail. For example, every countable tree is weakly homogeneous via principal measures, but in $L$ there is a countable tree whose projection is not determined. (There are also some less trivial examples.) On the other hand, every homogeneous tree is determined, and a proof of this is implicit in Martin's proof of $\Pi^1_1$ determinacy from sharps as mentioned in Andres's answer below. $\endgroup$
    – Trevor Wilson
    Commented Jun 17, 2013 at 19:23

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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 It is an unpublished note; the latest version is dated May 29, 2008, and can be downloaded from his page.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.

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