I don't see immediately how to use a hint specifically about top dimensional forms to deal with something related to $1$-forms (unless it was intended for you to deal with $1$-dimensional submanifolds $N$ of $M$, but you would have to prove something like: if $\int_N \omega = 0$ for every submanifold $N$, then $\omega$ is exact. This can be a bit of a hassle), so I'll allow myself to keep the core idea of that hint but to adapt it to the case of $1$-forms, while also trying to keep elementary. From here on, $\omega$ is a closed $1$-form in $M$.
Given any path $c: I \to M$ and a fixed point $p \in \pi^{-1}(c(0))$, you can lift it uniquely to the universal cover as a path $\widetilde{c}$ starting from such point $p$. It is trivial, by definition, that $\int_c \omega=\int_{\widetilde{c}} \pi^*\omega$. But the problem is that since we are aiming to gather conclusions about fundamental groups and loops, we want to handle loops, and the lift of a loop $c$ need not be a loop (of course, otherwise everything would be simply connected!).
But since there are finitely many points in $\pi^{-1}(c(0))$, we can concatenate the loop $c$ and start at where the lifted path ended each time, until some loop $l$ is formed. So, we get that $n(l)\int_c \omega = \int_l \pi^* \omega$, where we don't even know what $n(l)$ is (only that it is a non-negative integer), but it doesn't matter: since $l$ is a loop and the universal cover is simply connected, this implies that the right side is zero, and thus $\int_c \omega=0$.
But this holds for every $c$. Thus, $\omega$ is exact. Note that this also disregards orientability altogether, which is not necessary for the result to hold.
There are plenty of other ways to see this. The link provided by Arnaud in the comments of his answer is one of them. For yet another way, we can argue algebraically, since this is immediate from the Hurewicz theorem, the characterization of finite abelian groups, the universal coefficients theorem and the fact that deRham cohomology is isomorphic to real-valued singular cohomology.