A set is countable means there exist a enumeration procedure to generate each of its elements and for a given element of set, it take finite step to generate it using the enumeration procedure.
Let $∑ = {a,b }$ and there exist a enumeration procedure to generate all the string of the language $∑$$^{*}$.
$∑$$^{*}$.$=$ ${ ∈$ , a , b , aa, ab, ba, bb , aaa , ........ }$
Here enumeration procedure is simply the generating string of the language by length for the fixed length string are in alphabetical order.
This way $∑$$^{*}$. is countably infinite & $2^{\sum ^{*}}$will be uncountable set
Because the power set of countably infinite set are uncountable.
Ref: http://www.cs.xu.edu/csci250/06s/Theorems/powerSetuncountable.pdf