How do we know that water has [H+] = 10^-7?
Why is $[\ce{H+}] = [\ce{OH-}]$?


Because the water is electrically neutral.  

In general, for all solutions, the sum of the concentration of each ion, multiplied by the ion's respective charge, must be zero. 


  How do we know $[\ce{H+}] = 10^{-7}$?


This is found by experimentally by determining $K_\mathrm{w}$ and substituting, or by measuring the pH.  

For details see: Bandura, A. V.; Lvov, S. N. The Ionization Constant of Water over Wide Ranges of Temperature and Density. J. Phys. Chem. Ref. Data 2006, 35 (1), 15–30. DOI: 10.1063/1.1928231.