Clarification about dual space of a infinite-basis vector space

Question

I've got this excercise:

Let $V$ be a vector space with basis $\{v_{i} \}_{i \in I}$. For each $i \in I$, let $v^*_i \in V^{\ast}$ defined by $v^*_i (v_j ) = \delta _{ij}$

Show that $\{v^*_i \}_{i\in I } $ generates $V^{\ast}$ if only if $I$ is finite.

I've read this answer but I can't understand why this is a proof. What are we doing when we define the two disjoint finite sets? How can I "show that $f$ works"?