How do I use generating functions?

Question

So, I have been taking an introductory combinatorics class and over the past semester we had, not surprisingly, occasionally used generating functions to solve problems. Now, I am capable of adequately "solving" problems relating to generating functions, but I am still confused on something a bit more fundamental, namely, how these functions are useful. For example, if I apply a generating functions to answer a question about, say, $k$ partitions of $n$, and I wind up with a formula in terms of $x$ (like $1/(1+x^k)$ or something like that), I don't understand how that then helps me solve the original problem about $n$ and $k$, since $x$ can be chosen arbitrarily, as long as it is less than $1$. The only time I've been able to see how I can use generating functions to solve problems directly are when I've applied them to things like recursion. I know this probably sounds like a super basic questions, which it is, but any explanation is appreciated.