Ha I've been learning the piano for over 30 years and I still feel the same as that quote and am not convinced I can hear so many voices at once so well, though admittedly I have improved a little recently because a lot of the music I have been playing has three or more voices and to play them you must hear them.

So playing an instrument (usually keyboard) helps - JSBach's inventions and sinfonias were written to teach counterpoint IIRC. If playing is out of scope then I found listening to orchestral music can be helpful because you can latch onto different instruments and learn to interpret what they're doing individually (first) then how they are interacting with other voices after. Strong quartets will similarly allow you to listen to multi-voice music that may sound more distinct than just keyboard.

The other piece of advice is to get very familiar with a particular musical work and listen to it repeatedly. Once the familiarity is there you'll start to pick up different things on repeated listens. If you then start changing the recording/performer you'll also notice how they choose to interpret passages differently and certain voices may sound more prominent which can be a hook for your ear to latch onto...

Just some initial thoughts, hope that helps!

EDIT - Pachelbel's Canon in D is probably one of the most accessible songs to practise listening to. Poor cellists always grumble about playing this one, see if you can hear why