Watts are the energy required to a move a mass a certain distance in a known time period. So in bike speak, the mass is you plus your bike, and the distance is the ground covered.
Moving a bike, though, is a far more complicated scenario, as its resistance to motion is far from consistent. In layman's terms, then, this equates as: power = force x velocity.
And that is the key to understanding how a power meter works. It's essentially applying that equation to a given part of the bike - be that bottom bracket axle, crank, hub, pedal axle etc.
Accurate measurement of this force is one of the biggest challenges power meter manufacturers face, placing the upmost importance on the smallest of details such as the placement of the gauges, the quality of the gauges themselves and even the temperature of the measured material.