... 'at which the tuned amplifying circuits can operate with maximum stability, selectivity and sensitivity'.
That rather woolly phrase is correct, but lacks the emphasis to direct you to what is happening.
The input frequency is converted to a fixed frequency, and then filtered. Further amplification and filtering takes place at this frequency.
The input signal does not enter the radio by itself, it has potentially dozens of other, often stronger, signals at nearby frequencies. These must be filtered out before too much amplification takes place, otherwise they will overload the amplifiers and contaminate the wanted signal.
It is difficult enough to build a good filter, even more difficult to build a good filter that can be tuned over the whole range of frequencies that the radio is to receive. Better to just build one good filtered amplifier at a lower intermediate frequency, and convert the wanted input frequency down to that.