I am trying the get a deeper understanding of the following lemma:
Let $V \subset \mathbb{R}^{n+1}$ be open; and let futhermore $U \subset \mathbb{R}^{n}$ be open with $U \times [0,1] \subset V$. We also define $i_t: U \rightarrow V$ by $x \mapsto (x,t)$ for $t \in [0,1]$. By $\Omega^k(V)$ we denote the space of differential k-forms on V. Then for each closed $\varphi \in \Omega^k(V)$ there exists a $\psi \in \Omega^k(U)$ such that $$i_1^* \varphi - i_0^* \varphi =d\psi $$
This lemma was originally introduced as a mere technical construction to package away the heavy lifting of the proof of the Poincarè-lemma, but we have used it multiple times since and i came around to taking it as a quite deep result, since it of course yields that pullbacks of closed forms along homotopic functions only differ by an exact form, thereby establishing a connection between forms and notions from topology.
I would therefore like to get a bigger picture of what is happening here, why the above lemma should be true morally. I feel that I'm not quite grokking what is going on.
I skimmed through wikipedia and a couple of textbooks and came across de Rham-Groups, which we did not cover in our lecture, but that seem to have something to do with this (because you look at forms modulo the exact ones). I also tried throwing Stokes Theorem at the last equation, but it did not yield anything illuminating.
As a post-script, answers like "the only way to get intuition on this is to wait for an actual differential geometry/algebraic topology class" would be fine, as well as links or just some key-words to look up.
Thanks!