This is a rather long story that basically spans most of Earth's history. In order to understand where the continents came from we have to go back to about the beginning. As our solar system formed rocky and metallic asteroid began to collect in the part of the solar nebula to for Earth. As these bodies came together to form our planet, the heat of their impacts eventually cause Earth's surface to melt. This created a magma ocean. As this magma ocean began to cool Earth developed a crust that was probably made mostly of rock that were similar in composition to basaltic lava. (At this stage in its history, Earth probably looked very much like the planet in Star Wars: Revenge of the Sith where Obiwan and Darth Vader had their climactic battle.)
As this crust and the underlying mantle cooled we eventually say the formation of the lithosphere. (This is a layer of the Earth that contains all of the crust and about the upper 100 km of the mantle.) As this layer cooled it got denser. Eventually it got dense enough that it could sink back into the mantle. By this time the ocean had probably also formed. So, this sinking slab of lithosphere would have also absorbed a significant amount of water. As the slab sinks, it is heated until the water begins to come off of it and flow up into the wedge of mantle that overlies the sinking plate. This water reduces the melting point of the rocks causing the to partially melt.
Now, one of the interesting results partially melting rocks is that you end up with a magma that has a slightly different composition than the parent rock. So, the magmas created at these subduction zones is somewhat higher in silica and aluminum than the rocks for which they are derived. This magma works it way upward eventually creating volcanism on the surface that is basaltic to andesitic in composition. (We can see this process at work to day in the Aleutian Islands off the west coast of Alaska.) Now we think that this is the first stage in the creation of continental crust. It is not quite there yet, but it is the first stage.
As this process continued these mostly andesitic volcanic islands grew bigger and bigger. Eventually as the processes of plate tectonics continues some the island arcs may have crashed into each other forming even bigger land masses. Others may have just grown large on their own. Either way, the island crust probably grew too big for the basaltic magmas generated in the underlying subduction zones to rise through. So, these magmas started heating this crust causing it to begin to melt. The partial melting of the andesitic crust produced magmas that were andesitic to granitic in composition. These magmas were able to penetrate and create an new type of crust. This was the first true continental crust.
From here the continental masses just kept grown bigger and bigger. As the masses grew they were carried about Earth's surface as the processes of plate tectonics kept constantly rearranging things. As they moved about Earths surface, the continents kept colliding with each other and then breaking apart again. By about 300 million years ago (mega-anum or Ma), most, if not all of the continental masses had come together to create the supercontinent that we call Pangea. (There had been other supercontinents before this point, but they aren't really important to this question.)
Then about 200 Ma Pangea began to break apart again. This break up occurred in three different phases that resulted in seven different continental masses. These were North and South America, Eurasia, Africa, Australia, Antarctica, and India. (Note that the division of Eurasia into the "separate" continents of Europe and Asia is more of a geopolitical/cultural division rather than a geological division as it is with the other continents.) From these positions, plate movements carried the continental masses to their current position. India came all the way up from Antarctica to eventually collide with Asia.
This in a nutshell is how the six continent formed. There is a lot in this explanation that may need further explaining for some people to understand. The problem with historical geology (which is what this question is about) is that you first need to understand physical geology. Physical geology is the study of the processes that shape the Earth. It is only after one understands these processes that one can really begin to understand what happened in the past. This is why, in a geological education one must first study physical geology before they start to study Earth's history. So, if there are terms and concepts in this answer that you don't understand, then I suggest doing some research on them. Once you do understand them, this answer will make much more sense.