And it's so singable. In our 2006 production, all the guys who had principal roles were dying to get to sing it, and they'd occasionally jump in during rehearsals. It's easy to take a perfectly designed melody like this for granted. Probably the most distinctive feature is that the men begin in a high register on a pitch one step above the tonic, which gives it a little extra swagger. (If this doesn't make sense to you, imagine starting "Joy to the World" one pitch higher, but still in the same key. Listen.) The effect is almost as if the exaggerated motion of swinging the axe back causes them to overshoot the pitch.
Maybe the woodcutters are just showing off for the country maidens who are flirtatiously watching. In fact, the country maidens take over for most of the rest of the chorus (not included below) and then, in one of those basic but brilliant compositional strokes, the maidens' melody is combined with the lumberjack tune, blending harmoniously and yet each sounding a distinctive character. It's so thrilling, I included that section as bonus at the end of the video below.
I've taken to calling this chorus "The Lumberjack Song," even though the Monty Python gang owns that name. One of the purposes of these karaoke bits is to reveal how many great tunes there are in the show, so it's worth remembering that Gounod is probably best-known for a great melody (Ave Maria) he crafted above a Bach prelude. Since there is an equally famous Schubert setting, we generally speak of the "Bach-Gounod Ave Maria," so let's call this "The Gounod Lumberjack Song." Just follow the bouncing axe!