In the derivation of $K_\mathrm{w}$, the ionic product of water, it is stated that
$$[\ce{H+}][\ce{OH-}] = K_\mathrm{w}$$
and substituting $[\ce{H+}] = [\ce{OH-}] = 10^{-7}$ we can find that $K_\mathrm{w} = 10^{-14}$.
How do we know, though, that $[\ce{H+}] = 10^{-7}$? And why is $[\ce{H+}] = [\ce{OH-}]$?