Where is fuel metered in continuous flow fuel injection system?
Just in front of the engine. It makes most sense to measure fuel flow as close to the engine as possible: any leaks between meter and engine will distort the measurement. Here is an example of the B737 arrangement.



As to the how, airborne fuel flow meters work in the same way that turbines do: the mass flow of the fuel causes an impeller to spin, rotational speed is then a measure of fuel flow. Note that it is a measure of fuel mass flow, not fuel volume flow: mass flow is not subject to variations due to temperature etc.