Here's my understanding. Ammonia is produced as a byproduct of fish waste. In a plantless tank, one set of bacteria convert this to nitrite, then another set of bacteria convert it to nitrate. Since nothing is in the tank to use of the nitrate, it builds up awaiting a water change.
Now, plants need nitrogen. I believe the ammonia form of nitrogen is one of the easiest for plants to use. When the easier ammonia-form of nitrogen is gone, the plants go to work on the NO3 variety. Thus, with sufficient plants, you don't really need to go through the "normal" tank cycling (which is just building up the bacteria base) because the plants can assist in the process. That's also why in planted tanks, the filter is primarily needed for mechanical filtration and water movement. With the extra surface area in the tank caused by the plant mass, the tank isn't as dependent upon the surface area in the filter sponges, bio-products, etc. And, as you don't want to strip out lots of the stuff in the water with plants, carbon material is considered a negative in a heavily planted tank.