***** NOT A TEACHER *****
Most books seem to classify "that" in your sentence as a subordinating conjunction.
A few books do not think that it deserves the dignity of being labeled a conjunction, for it does nothing but introduce the noun clause. So those few books use the word expletive. But even those books admit that nowadays it is simply called a subordinating conjunction.
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You may like this extra information.
Many, MANY years ago in England , the "that" WAS necessary.
According to scholars, your sentence might have been analyzed like this:
"All I know is THAT: He fell down the stairs."
"That" was a demonstrative pronoun (when you point with your finger). "He fell down the stairs" explained "that" (the idea that you were pointing to).
Over the years, the English people started to think that the word "that" ONLY introduced the idea "He fell down the stairs," so they started to drop the word "that."
Source: House and Harman, Descriptive English Grammar (copyright 1931 and 1950).