I stumbled across this paper providing an intuition of differential forms. On page $3$ the paper reads
"Think of a vector as a pin and a one-form as an onion. You evaluate a one-form on a vector by counting how many onion layers it goes through. More generally we represent a one-form on an $n$-manifold by drawing $(n − 1)$-dimensional surfaces on it. We call these surfaces leaves. In the general case we can still count how many of these leaves each vector goes through. [...] This picture makes it easy to integrate a one-form."
I understand the definition of a $k$-form, as well as this (different?) interpretation of them, yet the phrase "You evaluate a one-form on a vector by counting how many onion layers it goes through" remains a mystery to me.
What does the quoted passage mean?
If your differential form is $dx$, and you evaluate it on the vector $c \ \partial_x$, you get $c$, because it "pierces through $c$ depth of the onion $dx$". $dx$ evalutes to $0$ on other directions, so if I add some component $a \ \partial_y,$ the vector gets longer but I don't pierce more onion of $dx$, I just stuck the toothpick in on an angle.