MOSFET current mirrors
Prerequisite knowledge: Understand the three terminals of a MOSFET transistor (Gate, Drain, and Source) and the equations for the three modes of operation (Cutoff, Linear, and Saturation). You can find all of this on Wikipedia.

In words:

First, note that there is only a single path for the current \$I_1\$ to flow through the circuit (through M1 and M0 to ground). This is because zero current flows into the gate of a MOSFET. Thus, we know that the drain current (\$I_D\$) of M0 and M1 are equal.

Second, note that M0 and M1 must both be operating in the saturation region because their gates are tied to their drains (\$V_G = V_D\$). This means that the drain current of each transistor is determined by the square of (\$V_{GS} - V_{Th}\$). Because the two transistors are identical, they must have the same \$V_{GS}\$.

In equations:

\$I_{D_0} = I_{D_1}\$

\$\frac{1}{2}\mu C_{ox}(V_{GS0} - V_{Th})^2 = \frac{1}{2}\mu C_{ox}(V_{GS1} - V_{Th})^2\$

Assuming the two transistors have identical \$\mu\$, \$C_{ox}\$, and \$V_{Th}\$:

\$V_{GS0} = V_{GS1}\$