Q:

Why can’t you say “I fell the stairs”?

The verb to fall strongly implies the direction down, but in some circumstances it is obligatory (in StdAmEng) to include the word “down.”  The example I have in mind right now is

I fall down the stairs
  * I fall the stairs

Formally speaking, what’s going on here?  Note that this is not the same as

I fall [down] [to/on/...] the ground
  * I fall [down] the ground

Here down is optional (it sounds a bit redundant to my ear, but neither option is wrong), and [preposition] the ground is a normal prepositional phrase.  In other words, I don’t think down is acting as a preposition in “I fall down the stairs.”

A:

In English, an intransitive verb is a verb that does not take a direct object.  "To fall" is quite simply an intransitive verb.  That's pretty much all there is to it.  
Other intransitive verbs include "to go", "to happen" and "to run" (in the sense of quickly moving, as opposed to "to run" in the sense of operating a machine, or an organization).  It is not correct to say "I go market" because a preposition is required: "I go to market".  "I fell to the ground."
ETA: I'm racking my brain, Zack, trying to figure out how to give a more "complete" answer.  It is a matter of basic grammar.  Intransitivity means that the verb does not take a direct object.  That means that if it has an object at all, it must be indirect.  That's why the preposition is required: it signifies an indirect object.  It happens that many intransitive verbs pertain to motion, and for that reason the indirect object's prepositions tend to those that pertain to motion: to; up; down; and so forth.
Can you "fall the duck"?  Can you "run the sidewalk"?  That's what it would take to omit the preposition and pretend the verbs "fall" and "run" were transitive.  Do you not see that there is nothing further to say about it?

A:

to fall down is a bipartite verb. Loads of English verbs are bipartite, including:

to throw out
to pick up
to fill in 
to see through
to figure out
to show off
to go away
to write up

These verbs all have distinct senses compared to the plain verbs.
To fall and to fall down have very similar meanings, but they are distinct. As outis nihil noted, 'to fall down the stairs' is very different to 'to fall on the stairs'. On the stairs is a locative adjunct, it is an optional phrase telling you where the action happened. To fall means that someone or thing that was standing collapsed under gravity. To fall down the stairs means that someone collapsed and tumbled down a stair case - they didn't collapse and stay on a single step. You can also fall down escalators and hills and Egyptian pyramids, anything you can tumble down! So while to fall is intransitive, to fall down is transitive.
If you're interested in reading more about bipartite verbs this is a good article (and where I got the examples from): The bipartite structure of verbs cross-linguistically, or Why Mary can't "exhibit John her paintings" by Heidi Harley.

