The proof of the product rule for derivatives in a book I have goes like this:
$$\begin{align} (f \cdot g) (a) &= \lim_{h \to 0} \frac{ (f \cdot g)(a + h) - (f \cdot g)(a) }{ h } \\ &= \lim_{h \to 0} \frac{ f(a + h)g(a + h) - f(a)g(a) }{ h } \\ &= \lim_{h \to 0} \left( \frac{ f(a + h)[g(a +h) - g(a)] }{h} + \frac{ [f(a + h) - f(a) ] g(a) }{h} \right) \\ &= ... \end{align}$$
I can't get the second to the third step. What exactly happened in the numerator?
I mean it's a correct manipulation, but I would like to understand what kind of manipulation was done there.
In analysis (in particular, in calculus proofs involving limits), it is very common to rewrite a difference $A-B$ as $$A-B=(A-C)+(C-B),$$ which is obviously true.
In your case, this trick was applied with $$A=f(a+h)g(a+h),\quad B=f(a)g(a),\quad C=f \left( a + h \right) g \left( a \right).$$
In fact, is also very common to rewrite a difference $AD-BE$ as $$AD-BE=(AD-AE)+(AE-BE)=A(D-E)+E(A-B)$$ in order to apply known informations about $D-E$ and $A-B$ (which is exactly what happened in your case).