Why is a set of attributes X not a key if X → Y holds but there exists a strict subset V of X where V → Y?
Generally, you want to define a key as a superkey that is minimal with respect to inclusion; if $X$ is a superkey, $V$ is a superkey and $V \subset X$, then $X$ can't be minimal.

The intuitive idea behind the definition is that we don't want keys to contain "useless" attributes. For instance, imagine a table that contains a list of people, each of which has a unique number $K$, a name $N$ and a surname $S$. $K$ would be a key to that table, because it functionally determines all other attributes in the relation. The pair of attributes $\{K, S\}$ also functionally determines the entire relation, but attribute $S$ is "useless" in that respect, because it is itself determined by $K$.