Why are all finite languages regular?
The pumping lemma states that if $L$ is a regular language, then there is an integer $p \geq 1$ (the "pumping length") such that every string $w$ in $L$ of length at least $p$ can be written as $w = xyz$ satisfying (1) $|y| \geq 1$, (2) $|xy| \leq p$ and (3) $(\forall n \geq 0)[xy^nz \in L]$.  (Source: https://en.wikipedia.org/wiki/Pumping_lemma_for_regular_languages#Formal_statement)

If $L$ is finite, then one can take the pumping length $p$ to be any integer greater than the length of any string in $L$.  The three conditions above would then be satisfied vacuously because there is no word in $L$ of length at least $p$.