Prove that for a n-bit word $x$ the operation
$$x \& (x - 1)$$
with $\&$ being the bitwise AND-operator, turns off (inverts) the rightmost bit of $x$ (e.g. 0101100 -> 01010000). The example above is is the first bit manipulation presented in Henry Warren's Hacker's Delight and I would like to understand why this works.
A concrete mathematical proof or even pointers to a general strategy when undertaking proofs regarding bitstrings would be higly appreciated.