I believe they try and shift the phase slightly. If the mains in present, it will overwhelm the inverter and keep it locked, or at least show up as a distorted waveform. If it can swing the phase freely, then there is no mains power, so it shuts down.

E.g. (in Europe) an oscillator with a frequency of 49.95 will be dragged to the mains 50.00. By measuring the frequency it will read 50.00, if it drifts low, then it is not phased locked, and so will shut itself down till 50hz reappears.

Think of it like how metronomes come into harmony. Only 1 is a little desktop unit (your inverter) and the other is building sized (the grid).