What method do wireless routers use to break apart the received wave into different frequencies?
The signal from the antenna is limited to the desired RF band(s) using SAW filters, so that strong non-ISM signals (e.g. DECT phones) can not upset the mixer.

Channel selection is typically performed as a classic heterodyne receiver using a PLL as clock source for the LO, and another SAW band-pass filter on the IF, or a simple low-pass filter in a zero-IF design.

The thousands of carriers used in OFDM are split digitally in the base band processor using FFT.