All living things require 6 elements to surivive. These elements are Carbon, Hydrogen, Oxygen, Nitrogen, Phosphorous, and Sulfur (CHONPS). Plants obtain these nutrients either through the air or by sucking them up through their roots. Once the nutrients are incorporated into the plants they are no longer avaliable to other plants. Death of these plants causes the nutrients to be released back into the environment so that other organisms end up with an abundance of them (so it is basically nature's fertilizer). Fire speeds up the process of returning these nutrients to the environment. This allows new plants to grow rapidly and densely.
It's not the same vegetation that grows so thick and fast, at least at first. When the ground is cleared by fire, smaller plants can then get the sun they need to grow--sun that was blocked by larger plants, before the fire. This is why there are different trees that characterize different age forests; at first smaller plants, shrubs and trees can grow. After a while, the new types of trees that sprout are those that can grow when the first plants are providing some shade.
Also, after plants are burned, in some cases the ashes that are added to the soil make it more fertile.