When the computer doesn't get a signal in the appropiate range from the O2 sensors, it stays in a safe, or limp home mode, and turns on the check engine light. In the safe mode the computer just uses a set of safe parameters for fuel and timing and doesn't adapt, which means it runs rich and with little timing.
Notice how when you first start your car, its a little slugish until the tempature gets up? O2 sensors don't work until they get hot, so the computer goes into a similiar setup called an open loop. When the O2 sensors start working the car switches into closed loop mode where it uses the feedback from the sensors in order to attempt to optomise running.
The after market stuff just sends the computer a "everything is just fine" signal, some voltage, to fool the computer into thinking the O2 sensors are working.
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