Let $T:V \mapsto W$ be a linear transformation. We define the dual transformation
$$T^*:\mathcal{L}^k(W) \mapsto \mathcal{L}^k(V)$$ (which goes in the opposite direction) as follows:
If $f$ is in $\mathcal{L}^k(W)$, and if $v_1, ..., v_k$ are vectors in $V$ then
$$(T^*f)(v_1,...,v_k) = (f(T(v_1)),...,f(T(v_k))$$
This is a section from Munkres-Analysis on Manifolds (Differential forms pg-$224$).
This seems meaningless to me and unnecessary to me.Can someone explain to me what exactly is the use of this definition or what exactly is this trying to explain?