Deductive proof - need help, explanation how to

There are many styles of natural deduction, and the one most suited for practical use is Fitch-style, which uses indentation just like programming languages to denote scoping. Basically, you ensure that every sentence you write is true in its context, where the context is captured by headers exactly as in a multi-level list. We can even throw away the lines at the side. See here and here and here for some examples.

Here is what such a proof of your example will look like: $\def\imp{\rightarrow}$

If $\forall x \ ( P(x) \imp Q(x) \land S(x) ) \land \forall x \ ( P(x) \land R(x) )$:

  $\forall x \ ( P(x) \imp Q(x) \land S(x) )$.

  $\forall x \ ( P(x) \land R(x) )$.

  Given any $x$:

    $P(x) \imp Q(x) \land S(x)$.

    $P(x) \land R(x)$.

    $P(x)$.

    $R(x)$.

    $Q(x) \land S(x)$.

    $S(x)$.

    $R(x) \land S(x)$.

  $\forall x \ ( R(x) \land S(x) )$.

$\forall x \ ( P(x) \imp Q(x) \land S(x) ) \land \forall x \ ( P(x) \land R(x) ) \imp \forall x \ ( R(x) \land S(x) )$.

I'm confident you can figure out how each line follows from the preceding ones. That ultimately is the goal of natural deduction, to make the logical reasoning clear.