The harmonic patterns, specially in popular music, are quite limited, once you get familiar with the usual ones, (and know the basics of tonal music, of course), it's usually easy; even for those people (as myself) which has very little ear... Always speaking of popular simple standard music, as this one.
I usually first play the main notes of the melody (perhaps the bass if it's very audible), to make a first guess of the scale, then listen the phrases, the timing of the cadences, and quickly find the tonic/dominant. In this case, it's easy to find the dominant (the "tension" chord, which asks to fall into the stable -tonic), for example, in 1:50, or 2:27. It's a E major chord (we verify it's working as dominant by adding the 7th... yes, it fits). The tonic then must be A ... major or minor? Clearly, minor.
From then, its easy to identifiy the following chords: I hear this (very standard) progression
Am Dm G7 C (A7) Dm Am Am/G F E(7sus4)
(this spans, for example, 2:30 to 3:06)