The Grove Electricity sensor is a current transformer able to give your project the ability to measure up to 5A AC current.
A fluctuating AC current will induce a similar AC current in an adjacent wire. Aren't electronics fundamentals wonderful? In this case, pass one wire conducting AC power through the middle of this coil and plug the sensor into your Grove system. Now you can read the current consumption with your microcontroller.
Note: This is an AC current sensor. Sorry, no measuring the current on a DC / battery powered circuit with this!