That explanation is kind of oversimplified.

So old freezers were just a box with a coil inside that got cold. Often with a fan to insure the inside was uniformly cold. Frost free freezers are a box, with a coil, but the coil also acts to condense water. This is the complication -- the coil is cold when the compressor is running, and the water will freeze up. But when the compressor is off the coil is allowed to warm up enough for the water to melt, and drip down and out of the freezer. The compressor will cycle on and off to maintain the set temperature so the warming up bit was already happening in old freezers. The new part is that the coil is protected from the fan when warm so that it does warm up enough to melt the frost on it, and that when it drips, that water is channeled on warmish surface to the outside.