The process of firing clay pots is what makes them impervious to liquids and hardens them. The process has been undertaken since ancient times when people would put clay vessels straight into the fire and with the use of primitive technology like fire-stacks and bellows, they would maintain enough heat to bisque the clay.
Today our technology is much more reliable and we use gas or electric power to fire kilns. These sources are much steadier, and guarantee a smoother transition in temperature then a natural wood fire so it would seem that they are perfect options to go with. This could be true for many potters, but a portion still also choose to use primitive methods like wood firing. The reason lies in the unpredictability of raw flame. The fire and ash circling around in a wood burning kiln can not be controlled and therefore, it will do whatever it wants with the pots you fire.
The (debatable) downside of wood firing is that it can take much longer to do, slowly building up the temperature over the course of several days and in some cases over a week. This means that the kiln can not be left alone at any point during its firing time. to steadily raise the temperature the fire needs to be fed and stoked every few minutes. Usually wood firings will be undertaken by groups of people so each can take turns tending the kiln, and for those who tend to go with this method the experience is what they enjoy.