"Den lille pige kan ikke køre bil."

I have these sentences: "Den lille pige kan ikke køre bil." and "The young girl does not know how to drive."

A more direct translation of the Danish sentence would go like this: "The little girl is unable to drive a car". Strictly speaking she might actually know how, but something (size perhaps - or her mother) is preventing her. She might be able to drive heavy machinery. But it is taken to mean that she does not know how do drive in general.

You ask why the sentence does not need an article. My answer may annoy you. "Because it doesn't." It's just the way Danes speak. "Den lille pige kan ikke køre en bil" is actually okay, sort of. It's just not idiomatic. I am sorry that I do not have something more substantial to offer, but questions beginning with "why" are notoriously hard for me to answer.

I suppose leaving out the article makes it "cars in general" rather than "one car". Logically speaking it should be plural then; "Den lille pige kan ikke køre biler" / "The little girl does not know how to drive cars", which is also sort of okay in both languages, but not really idiomatic in either.

Perhaps it has something to do with the origin of the word "bil". "bil" comes from "automobil". Danes cannot be arsed to say "automobil", so they leave out "automo" and are left with "bil". The same thing goes for "bus" really. Germans went the other way and kept "auto" instead (for cars at least).