The FM slope detector relies on the signal entering the input terminals (on the left) to be current in nature. This can be achieved by placing it in the collector of a common-emitter stage but can also work when feeding it from a voltage source via a high value resistor. If the transformer has fairly low coupling between primary and secondary this can also work with a voltage drive at the input.
The peak parallel tuning of the circuit is offset from the FM carrier frequency (usually an I.F. of 10.7 MHz) by several hundred kHz hence, the circuit output amplitude rises as the modulated carrier gets closer to peak tuning and falls as the modulated carrier gets further away from the peak tuning point: -
Picture from here.
This means that the carrier is regarded as being placed on the slope of the parallel resonant tuning and hence, is called a slope detector. It converts a flat-amplitude and modulated FM carrier to a modulated carrier where the frequency modulation effects are transferred to the amplitude and are resolved with a conventional diode detector (as shown).