Why does an injection from a set to a countable set imply that set is countable?

Unfortunately, there is no uniform agreement to the meaning of "countable". Specifically, does it mean only countably infinite, or do we include also finite sets?

Well. The answer depends on context, convenience, and author. Sometimes it's easier to separate the finite and infinite, and sometimes it's clearer if we lump them together.

If $B$ is countable denote it $B = \{b_n\}_{n \in \mathbb{N}}$.

If $f : A \to B$ is injective for all $a \in A$ there is a $b_n = a$. Since the map is injective two different elements in $A$ map to different points in $B$, so can you see how to enumerate $A$ now?