Fama got the prize for his "efficient markets" theory, and Schiller shared it for showing that Fama was wrong.
That pretty much captures it. (As for Hansen, the third recipient of the prize, hardly anyone seems to understand his work, and those who do seem incapable of explaining it in ordinary English.)
Actually, Fama and Schiller ended up in the same place but for diametrically opposed reasons. Both believed no one could predict future moves in stock markets. For Fama, this is because markets possess and process way more information than any individual could possibly possess and process. For Schiller, markets are largely irrational (moved by what Adam Smith called "animal spirits"), and no one can predict what an irrational actor might do.
The 2008 financial crisis pretty much proved that Fama is wrong and Schiller is right.
UPDATE: John Kay adds that giving the Nobel Prize to Fama and Schiller together in the same year is like giving the physics prize in the same year to Ptolemy and Copernicus.