It is necessary for dishwashers to use a pump to drain because their sump -- the part of the machine that holds and circulates water -- is almost at ground level. The water has to be stored there so it is below the level of the lowest part of the door.
At the appropriate point in the dishwashing cycle, a solenoid -- an electromagnet -- activates the drain valve, which opens, and the pump comes on to expel the waste washing or rinsing water from the machine. The pressurized water is routed via a hose to a preexisting part of the building’s waste-water system; in regular homes, this is typically the drain from a nearby sink or a port specifically for this purpose in the garbage disposal.
Because dishwashers are invariably located on the floor close to an existing waste-water drain, they are below the counters. They drain through a hose attached to their lowest point, and they join into a waste-water system that is itself below the counters. At no point, therefore, does a dishwasher drain hose pass above the counters, so there is never a reason to drill a hole in the counters.
It is usually necessary to drill some holes through the side of the cabinet in which the sink is fitted, so that the dishwasher’s hot and cold feeds and its drain hose can be attached, and so that the electrical cord can be plugged in. In their online dishwasher tutorial, LG note that “the access hole for the drain hose should be 1.25 inches (32 millimeters).” These holes are typically low down in the cabinet side; they are made nowhere near the counter tops.