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The relevance of Shannon entropy is to repeated sampling: Given $n$ independent samples from a source with binary Shannon Entropy $k$, you can extract $nk(1+o(1)$ i.i.d. uniform bits as $n$ tends to infinity with probability tending to 1. This follows e.g. from the Keane-Smorodinsky [1] finitary isomorphism theorem. See also [2]-[5] below. [1] M. Keane and M....
It's the $\alpha^{\mathrm{th}}$ moment of the Tribus surprisal. This generalizes the statement that entropy = expected surprisal. Or in Ross's textbook, "expected surprise".