I have no clue how they work.
See the Wikipedia entry for a description of how it works. http://en.wikipedia.org/wiki/Current_transformer
The idea is that you'll get a current in the secondary that is (ideally) proportional to the primary current (i.e., that flowing through the center). You can convert this current to a voltage using a suitably-sized resistor. The center-tap construction of the current transformer lends itself to using two diodes to convert the resulting AC voltage to DC, filtered with a capacitor and sampled with the Arduino's ADC.
You'll probably want to get one of them and experiment with it. It is likely that the transfer curve (current to DC voltage) will be non-linear due to various magnetic losses. However, you can probably construct a piecewise linear approximation to the transfer curve that will work suitably well for your application.