I don't know whether you'll find this intuitive, but here is how I think about the Taylor expansion of the logarithm, which I will write as
$\log \frac{1}{1 - x} = \sum_{k \ge 1} \frac{x^k}{k}.$
(We get your identity by setting $x = \frac{1}{2}$.) This is equivalent to the identity
$\frac{1}{1 - x} = \sum_{n \ge 0} x^n = \exp \left( \sum_{k \ge 1} \frac{x^k}{k} \right)$
which admits the following combinatorial interpretation: the LHS is the exponential generating function of the number of permutations of a set of size $n$, and the RHS, as it turns out, is the generating function for the number of ways you can divide a set of size $n$ up into cycles - but this is exactly the cycle decomposition of a unique permutation!
This argument is explained in more detail in a series of posts on my blog starting here. It is a special case of the exponential formula in combinatorics and can also be proven (as I did in the posts above) using Pólya's enumeration theorem. You may also be interested to learn what this has to do with zeta functions.