A distance relay trips if the measured impedance of the circuit is less than the trip setting. Z=V/I. If a fuse blows, V=0, so the relay measures 0 ohms; the same impedance it would see for a bolted fault on the breaker terminals. The fuse failure relay prevents false trips, but it still has to let the distance relay work for an actual fault when the voltage is very low.
Some fuse failure relays compare voltages from two VT's on the same bus on the assumption that the same event won't blow both VT's fuses.
Many digitial relays detect fuse or VT failure by monitoring negative sequence voltages and currents. If it sees negative sequence voltaes but no negative sequence currents, it must be a fuse failure.
Another application is on generator voltage regulators to prevent overvoltage when the regulators sensing voltgage goes away. (Imagine unplugging the speed signal from your cruise control set at 70 MPH.) The fuse failure relay trips the AVR or puts it in manual mode.