Rephrasing your question, you are asking why is the conditional d.f. of a random vector given by partial derivatives. The answer is available, for example, [**here**][1].

Just as a side note: what you have on mind is a *conditioning*. Therefore you should write
$$
\mathbb{P}\left[U_{j}\le u_{j}|U_{1}=u_{1},\ldots, U_{j-1}=u_{j-1}\right]
$$
instead of
$$
\mathbb{P}\left[U_{j}\le u_{j},U_{1}=u_{1},\ldots, U_{j-1}=u_{j-1}\right]
$$
the latter being trivially $0$, given the uniformity of all marginals.


  [1]: http://en.wikipedia.org/wiki/Conditional_distribution