Timothy Burke at Cliopatria has sketched a near-comprehensive list of modes of historical explanation -- or as he calls it, different ways of saying "so what?" in history. Highlights:
4. The past is another country: our own times are made more particular by looking at just how different the past really was. Caroline Bynum, Holy Feast, Holy Fast; Robert Darnton, The Great Cat Massacre; Richard White, The Middle Ground.This is actually slightly different from explaining how historical argument works, and why we're persuaded by it -- although a few touch on that -- but does speak well to the problem of why we care about history in the first place.
5. The past helps us make N as big as possible: it is a source of data for making generalizations, formulating models, constructing claims about human universals. Jared Diamond, Guns, Germs and Steel; David Christian, Maps of Time...
16. The past is memorial: we study (recite it, really) it to honor what people did or sacrificed on our behalf. Tom Brokaw, The Greatest Generation.
I would add a few, especially if we're considering history writ large and not only what current academic historians practice. For example, we read and write history (especially biography) to find models of ethical behavior. (Hi, Robin.) Examples: Plutarch's Parallel Lives, many others.
On a slightly different tack, we read history to psychologize or better understand someone who is interesting for other reasons. Again, a big chunk of biography, especially of artists, celebrities, or political figures.
And we turn to history to win other arguments, whether factual or interpretive. If I think word X in poem Y by author Z means ___, then I might look at either Z's biography or some contemporary history to justify that claim.
Lastly, we use history to understand language, what words mean and have meant, where they come from and how they circulate.
I'm sure there are many more, but these are the first that come to mind.