The map $I$ you want to define has domain $(V/U)^*$ and codomain $V^*$.
You want to define its action on $g\in (V/U)^*$ (so $g$ is a linear form on $V/U$): $Ig$ should be a linear form on $V$ and therefore you need to define its action on every element of $V$. The natural choice is
Ig\colon v\mapsto g(v+U)
because you know that $g(v+U)\in F$ (where $F$ is the field of scalars).
Now you have to prove:
for every $g\in (V/U)^*$, $Ig\in V^*$ (that is, it is a linear map $Ig\colon V\to F$);
$I$ is linear.
Both statements are easy verifications.
There is a different way to see this. Let $T\colon V\to W$ be a linear map. Then we can define $T^*\colon W^*\to V^*$ by defining, for $g\in W^*$,
T^*g\colon v\mapsto g(Tv)
In other words, $T^*g=g\circ T$ (map composition), which spares you verifying point 1 above.
The particular case of the inflation is when you consider $T\colon V\to V/U$, the canonical projection.