The science of plate tectonics has explained the slow motion of the continents over time, since the Earth cooled and the first land masses appeared. Alfred Wegener (1880-1930) noticed that South America and Africa appeared to fit together, and proposed that at one time all the continents of Earth were contiguous. He named this super-continent Pangaea (meaning "all lands") which began to break apart into recognizable continents about 300 million years ago.
Because of the intense heat within the Earth, the matter which supports the continents is subject to current fluctuations and motions; these, in turn, subject the cooler land masses in the surface to continental drift. When continents crash into each other, subduction occurs, where one land mass is forced under another. The reverse process occurs where one land mass is pushed up against another and new land is formed.
Current computer models suggest that there have been several super-continents prior to Pangaea. Earth apparently has been recycling its land masses since it was formed. See more info (and some animations) at the link: