The probability generating function E[s^X] encodes a distribution; the moment generating function E[e^(tX)] encodes its moments.
The property that does the work: the generating function of a sum of independent variables is the product of their generating functions. Convolution becomes multiplication.
Interview uses: the distribution of a sum of dice, closed forms for recursively defined sequences, and deriving moments by differentiating at zero.
Rarely required, but producing one when a counting recursion resists direct solution reads very well.