Why do we refer to the floors of buildings as stories?

According to the Oxford English Dictionary, there has been a fair amount of debate on the exact etymology of the word storey or story. They write the following:

First in Anglo-Latin form historia; hence probably the same word as story n.1, though the development of sense is obscure.

Possibly historia as an architectural term may originally have denoted a tier of painted windows or of sculptures on the front of a building.

The current view that the word is < Old French *estoree ( < estorer to build, furnish: see store v.) is untenable on account of the Anglo-Latin form historia (from 12th cent.).

So story shares a link to historia, but even the OED is unsure how the current sense came about. Nonetheless, story has referred to a level of a building since around 1400.

I thought to quote some claims that I found via Google: Answer from [user] aaindian [the paragraph has been divided for readability]

(Medieval) In a painting of a building, each floor is a story.

from Medieval Latin historia, picture, story (probably from painted windows or sculpture on the front of buildings).

A century ago etymologists speculated that "story" came from some lost word "stairy," perhaps related to Gaelic staidhir, flight of stairs; or possibly from something along the lines of "stagery," derived from "stage." Others dismissed these as being obviously born of desperation, and for a time the experts settled on Old French estoree, a thing built.

But doubts arose when researchers dug up such phrases as una historia octo fenestrarum, "a story of eight windows," from medieval Latin history books. Historia in Roman times meant history or story, and by the Middle Ages had acquired the meaning of "picture." So the charming notion arose that medieval folk were in the habit of installing rows of windows in their buildings called "stories" that were decorated with paintings or sculpture. The theory is that these stories, which for all anybody knows may actually have told a story, eventually came to signify a level of a building.

Apparently as evidence of this practice, the authors of the Morris Dictionary of Word and Phrase Origins cite the fact that they once visited a Swiss-style hotel decorated along these lines in Lake Placid, New York. (Each floor was tricked out with a large hand-lettered slogan, such as "The only way to multiply happiness is to divide it.") At any rate, conjecture has now hardened into conviction. Believe at your own risk.

Sources: http://www.straightdope.com/classics/a2_249a.html

This Nov 20 2002 reply from Ken Greenwald via Word Wizard fails to clinch the etymology, but its minor details may help elucidate.

At the time when the usage came to be circa 1400 or 1500's, the general population was illiterate. Some religious began to draw biblical stories on the side of their homes. Many of them had structures with more than 2 floors, creating more than a single 'story'. When asked where they lived, they said the building with the stories. His room was on the second or third 'story'. I found that explanation easier to comprehend than the others. A picture is worth more than a thousand words, as it were.

I have the suspicion that German Stockwerk (storey) might have an influence. But this is only a first idea. But I would like to do some historical research about the word in Low German and Dutch. Semantically it is highly improbable that there is a connection with story and Latin historia. I suppose that the word had a special way of development with a lot of changes and finally came near "story" (for reading).

There is the German verb aufstocken, which means to build a new storey on a house. And a wordform such as *Stockerei might be related with storey. *stockerei - stogerei - sto••rei. This is only a play with letters, not the possible historical developmemt. But it shows that a relation might exist. I don't even know whether a word such as *Stockerei exists or existed somewhere. Perhaps Grimm has something.

A connection with Old French estorer (mentioned above) is possible. It belongs to the Latin word family instaurare/restaurare. Etymonline has OF estorer meaning erect, construct, build.

I can't find any older German word forms coming near *Stockerei. The idea of Stockwerk is connected with Stock (stick), forming the wooden construction as of a half-timbered house.