How do you make tapioca from cassava?
It is possible to do at home, but it is very labor intensive and requires equipment that most people don't have at home.  

The first major step is to produce tapioca starch (ie tapioca flour). The cassava must be cleaned and peeled, then finely grated or milled to break the cell walls and expose the starch.  This mass is then washed in a large amount of water and the insoluble pulp is strained out and the starchy water reserved.  The starch-water mixture, called "Starch milk", is allowed to settle for 6+ hours and the water is drained off until a thick slurry is left.  This slurry is then either purified further with more water then dried to form the flour.  The additional washing of the slurry is important to remove residual insoluble fiber and, more importantly, cyanide-related toxins naturally found in the cassava. Traditional cultures dried the slurry on basketwork trays, but it is more efficient to do it in an oven at low temperature with regular raking and turning. The dried starch would then be sifted and used for cooking.

Making tapioca balls or pearls from tapioca starch is relatively easy compared to the processing of the cassava.  The tapioca starch is mixed with boiling water to form a dough which is then kneaded and rolled into the desired size balls.  The tapioca balls are then cooked in boiling water until transparent.  

Other sources:

http://www.fao.org/livestock/agap/frg/AHPP95/95-81.pdf

http://phys.org/news90080234.html