How does this simple FM slope detector work?
The input goes through a tuned transformer with a bandpass response. This should be tuned so that the frequencies of the input signal is on either side of the slopes of the filter response, like in this examples where the input is at the right side of the peak:



The output will then be a varying amplitude depending on the frequency.



Afterwards, yes, then a slope detector is used to filter out the high frequency content like in an simple AM demodulator.