MOSFET Constant current - Current mirror circuit
How does the current not change through Q2 when the voltage at Q2
  increases?


Because the drain current, ID, is directly proportional to the gate-source voltage, VGS:
$$I_D = \frac{1}{2} \mu_n \cdot C_{ox} \cdot \frac{W}{L} \cdot (V_{GS} - V_{TH})$$

VTH is the gate-source threshold voltage and the others are geometry- and material-related parameters. Asssuming both MOSFETs are identical (i.e. having identical parameters), their drain currents will be equal since their gate-source voltages are equal (gates are shorted and sources are tied to ground).


  Where does the excess voltage go when Vcs is increased?


There's no excess voltage. If the drain voltage of the right-most MOSFET (labeled as VCS in the linked article) increases then the MOSFET will adjust its internal resistance so that the drain current remins the same (because still VGS is constant).