Why do we let the pork butt rest?

You should let any meat rest after cooking it. The juices retract into the center of the meat (the bulge in the middle which you see in some cuts) and if you cut into it immediately afterwards those juices will simply spill out. By letting it rest, the juices slowly redistribute throughout the meat so that when you cut into it more of the juices stay in the meat, making the meat more moist. Also, in the case of tough cuts like butts, resting in a fashion that they take along time to cool down (like wrapped and in an empty cooler to keep warm), then they hold at that finished temp longer helping to make it more tender without overcooking it.