Answer: All non-negative reals and negative integers.

Consider 3 cases.

1) $x \ge 0$. Then $|x| = x$, $[x] \ge 0$, and so $|[x]| = [x] = [|x|]$, which means that any non-negative $x$ is a solution of our equation.

2) $x$ is a negative integer. Then $[x] = x$, $[-x] = -x$, $|x| = -x$, and so $|[x]| = [-x] = (-x = |x| = [x]|)$, proving that these values of $x$ are also solutions.

3) $x$ is negative, but not an integer. In this case $|x| = -x$, $|[x]| < -x$. On the other hand, $[x] < x < 0$, $|[x]| > -x$, and the equality is not satisfied.