I know some people who can fall asleep pretty fast, but the term "fast asleep" doesn't seem to mean that. When I looked it up, all it showed was "someone sleeping tightly". How did fast come to be associated with sleeping?
The etymonline entry for fast says:
Old English fæst “firmly fixed, steadfast, secure, enclosed”, probably from Proto-Germanic fastuz, from PIE base past- (firm). The adverb meaning “quickly, swiftly” was perhaps in Old English, or from Old Norse fast, either way developing from the sense of “firmly, strongly, vigorously” (cf. to run hard means to run fast; also compare fast asleep)
It has a close meaning of “firmly, strongly” in “the box was strapped fast to the truck”, and as an adjective in “fast dye” (not fading) or “fast friend” (close).
In order to be 'fast asleep' the meaning of the word "fast" has to be defined.
"fast" (as already defined above) means "tight, safe, secure".
Hence, the phrase "fast asleep" basically means the same as 'sleep tight. But that still doesn't answer the question. So these two phrases mean the same thing, but what caused these phrases to come about?
Looking into history, the beds of last time were basically frames elevated from the ground with ropes tied in a kind of web underneath the mattress. In order to have a comfortable night, the ropes have to be tight. Hence, the expression,"Sleep tight!" And from that, the expression "fast asleep" was most probably derived.