The M79 was later replaced by the M203. The stock looks like that because it is generally fired from the armpit, rather than the shoulder. The M203 (which of courses attaches under an M16 or variant) is fired the same way at ranges greater than 100. You put the stock of the rifle in your armpit and basically elevate the rifle so the range indicator on the rear sight of the launcher lines up with the "nipple" on the front sight. Fire away and it makes a satisfying "donk!" noise. Accuracy is somewhat lucky guess and based in part on a reliable distance to the target and wind conditions. With an M203 I've bracketed targets at 300m, and I've been dead on at 100-150m. It is ineffective at ranges greater than 400m. I'm not sure about the M79, but the M203 sight is only marked up yo 400m.
I'm pretty sure the M79 operates the same way as the M203. It seems to be essentially a stand-alone 40mm grenade launcher. Fire from the shoulder at distances of <100m, and from the armpit at >100m. Yes, "thump gun" is an accurate description - when the round hits, there is a nice "thump" from the explosion.
Yes, you could probably put a round through a window at 300m with an M203. It depends on the wind, When I've fired HE at a 300m target, we had pretty horrible wind (80km'h gusts), but I still was hitting within a kill radius. Had there been no wind, I probably would have been dead on. The M79 should be no different.