"Fast" originally meant "secure", and is related to the German word "fest". The meaning of "quick" came later. According to Etymonline: "The meaning "quickly, swiftly, rapidly" was perhaps in Old English, certainly by c. 1200, probably from or developed under influence of Old Norse fast "firmly, fast." This sense developed, apparently in Scandinavian, from that of "firmly, strongly, vigorously" (to run hardmeans the same as to run fast; also compare fast asleep, also compare Old Norse drekka fast "to drink hard," telja fast "to give (someone) a severe lesson"). Or perhaps from the notion of a runner who "sticks" close to whatever he is chasing (compare Old Danish fast "much, swiftly, at once, near to, almost," and sense evolution of German "fix": "fast, fixed; fast, quick, nimble," from Latin "fixus")."

Oh! You asked WHY? Well, I think in Old English it was FAEST - which meant FIRM. But what I DON'T know is how it came to mean SPEEDY ! Anyone know?

Because many words that have come down to us from Old English (or before) have multiple meanings. In some cases, there were two different spellings for the two meanings and they've merged into one modern spelling.

Yes, it means he was thoroughly stuck in the trap. Couldn't get out. Hopefully a ship's anchor will hold it fast. (firmly, immovably, securely) Hopefully tonight I will be fast asleep. (not quickly asleep but deeply asleep!) If the colours of your orange Hawaiian shirt are not fast they will run and turn your whole wash orange or yellow. (not fixed/not properly fixed)