Why is the zero-th power of the empty set {ε}?
Strings with concatenation and the empty word form a monoid.

In monoids, the empty concatenation/sum/product yields the neutral element, i.e. $\varepsilon$/0/1.

An empty concatenation does not need any words to start with, so $\emptyset^0 = \{\varepsilon\}$. 

Intuitively, if you specialize


  $L^n$ is the set of all words obtained by $n$-fold concatenation of words in $L$ 


to


  $L^0$ is the set of all words obtained by zero concatenations (of words in $L$)


you can just stop reading after "concatenations" and the contents of $L$ do not matter; we don't even need it to be non-empty.