Hey, so this is a pretty cool etymology question.

The term "fast!" in archery is equivalent to the interjection "cease fire!" in shooting. This does indeed have a relationship to the term "hold fast," as the word "fast" in English originally meant "secure" or "firm" (as it still does today in some contexts). Hold fast is one example of this etymology in use, but we also see it in words like "steadfast" or even "fasten."

We believe the word originated from the Old English word "faest," meaning firmly fixed, or enclosed securely. You can see this word cognate in other Germanic languages like Dutch: "vast", meaning fixed or set, and German: "fest", same meaning. The term in English probably started to be used to mean "speedy" quite a long time ago, perhaps even in the Old English usage, deriving from the sense of "vigorously holding onto something" becoming just "vigorous" which became speedy. You can see this, for example, in something like "to run fast." Which meant "To run with a sense of sticking to it,"; then "to run with vigorous effort/without giving up"; then "to run quickly." Another good example can be found in the phrase "fast asleep," which means "locked into sleep," or "sleeping 'vigorously'", if that makes sense.

Now, returning to the archery usage. When firearms came about, they did not have self-contained firing mechanisms like modern weapons. Instead, it was necessary, early on, to ignite the powder or charge with a separate flame. Eventually, this flame would be attached to the gun (e.g., with matchlock weapons), but at the outset, and for quite a while with larger cannons, the fire was held separately. When people wanted the guns to discharge, they would call the artilleryman to set the light, or to light the fire. This became just "fire!" after a while.

In archery, obviously, there's no fire, so ordering archers to "Fire!" would not have been contemplated in the pre-gun era, and doesn't make sense in any era. Instead, archers would pull their bows back, tightly, and when it was time to fire, they would be ordered to "Loose!" their strings, i.e., let the strings go loose, thus launching the nocked arrows. Well, what's the opposite of "loose"? You guessed it: tight or secure. What's the word for tight or secure that they used? Yep, "fast." So, "fast" has been a term for keeping the bow strings held (either in a ready-to-shoot position, or just neutral) for a very long time, indeed.

Interestingly, this may, and I stress may, be the origin of the idiom "to play fast and loose," with something -- i.e., to mess around with something serious in a non-serious fashion, without regard to the consequences. This might have come from incompetent or irresponsible archery commanders constantly shouting orders to shoot and then cease fire, and then shoot, and then cease fire, as if they had no plan or didn't know what they were doing.

EDIT: I did a bit more research, and another theory for this usage (based on the same etymology above) may be that archers shouted it out to people near the range, warning them to "stand fast," i.e., not to get in between the archer and the target -- but this meaning doesn't seem to have as much historical backing as the primary, military one I used above, and should probably be considered apocryphal. Just thought I'd include it for completeness.