Why do people monitor the O2 sensor?

The oxygen sensor reading is one of the primary indications of the operating condition of the engine. People monitor it to try and make certain that their engine is getting adequate fuel. Too little fuel and engine damage can occur. See the ECU Primer for the details as to why this occurs.

Caution must be exercised by anybody that is depending on an A/F meter for engine tuning. There is evidence that the oxygen sensors in DSMs are not designed to be accurate except at their 'switch point' of roughly 500 mV. In other words, different sensors may give different A/F readings under the same conditions. There are also many other variables that can affect what is a 'safe' O2 reading for any given car.