It's an empirical fact. Electron density can be measured by various means (for example, X-Ray diffraction for the high-density regions). We can then do some quantum calculations and see: voila, if we evaluate the electron density operator with the wave function, it agrees with the measurements.
Note: The square of the N-electron wave function does not give the probability of finding an electron, but rather of finding a specific configuration of N electrons. E.g., if we have a wave function Psi(x1,x2) in space representation, then for concrete values x1 and x2, |Psi(x1,x2)|^2 is the probability of finding one electron at x1 and another one at x2 at the same time.