What is power and what will a power meter do to make me faster?

Without getting too science-y, power is the rate of work (or force times distance) a system is producing over time. That rate of work is represented in Watts which is Joules/Second or more simply energy expended per second.
A power meter, is a strain gauge and some electronics packed either in a crank set, pedals or the rear hub of a wheel. When you apply force to the pedals, the meter can sense disturbances in that force applied. (sorry, Obi Wan, couldn’t help myself) The meter also has a way to transmit that force reading to a handlebar display and capture device. The combination of these two devices measures the work being done and displays it in Watts instantaneously for the rider to see, in addition to capturing a data stream of power readings, usually in 1 second samples, of an entire workout for further analysis post exercise.
So in a simple but fuller picture, your body turns food into biochemical energy, and that biochemical energy is turned into mechanical energy when you turn the pedals. You are literally “doing work” by applying force to the pedals, producing power which is measured in Watts and in turn making speed! The meter is capturing a record of that output and displaying the immediate power output instantaneously.
None of this on its own makes a rider a faster. The meter and display unit are just giving you a number on your handlebar that changes every second depending on how “hard” or “soft” you are turning the pedals. The value comes in the interpretation of that data by a knowledgeable individual and then applying it to an athlete’s training program. The data captured by the meter provides insight into the athlete’s current ability and how hard or easy they are training at a given time, and enables the establishment of parameters to optimize how an athlete trains given their ability, goals, potential and individual situation. The less time and more constraints they have, the more beneficial this becomes to whomever is designing the athlete's training approach.
Next time we will go into a bit more detail why using power in training can be beneficial relative to other methods of measuring effort. If you have questions or comments, please feel free to post up.