There are a couple of ways to see why this holds.

Suppose \\(S\\) has \\(n\\) elements \\(x_1, \ldots, x_n\\). Let \\(A\\) be a subset of \\(S\\).
Then \\(A\\) can be represented by a vector of \\(n\\) bits, where the kth bit is 1 if \\(x_k\\) belongs to \\(A\\), or 0 otherwise.

So each subset \\(A\\) is represented by a unique n-digit sequence in base 2.

Since there are \\(2^n\\) such sequences, it follows that power set has size \\(2^n\\).