Learn English – “Hair” vs “hairs”

The reason I like to ride a bicycle doesn't work is because the singular indefinite article implies there's just one particular bicycle you like riding.

When you say I like to ride bicycles there's an implied plural indefinite article there, but we discarded that word centuries ago. We're so used to not using it we don't even notice it's not there.

Even though the article is missing, we understand as if it were there; we don't suppose OP likes riding more than one bicycle at a time - he likes riding [some, most, many, all, any] bicycles.

I think the success of this usage encourages us to discard the singular definite article in various related contexts. So I like to play the piano, is increasingly giving way to I like to play piano (even more noticeably with the more modern I like to play guitar).

This particular it is a Dummy Subject pronoun, Distance it; the construction requires a locative of some sort and estimates the extent of some stretch of (perhaps metaphorical) landscape.

• It's 31 miles as the crow flies from Bellingham to Mt. Baker.
• It's a long way to Tipperary.
• It's just corn out there, as far as the eye can see.

In the quoted sentence (quite a famous one in linguistics, because Haj Ross used it as the Fragestellung (pp iv-v) of his extremely influential dissertation, though he blames it on William James, not Bertrand Russell)

• It's turtles all the way down.

it doesn't refer to turtles, but rather — if it can be said to have any reference — to the "landscape" metaphorized by the old lady's comments (interestingly, it's always a little old lady in the story, though the prominent intellectual varies).

That is, we are seeing, through her imagination, turtles stacked one atop another below us, for as far as we can see (using see metaphorically, of course, but that's normal for imaginary landscapes).