To understand this, you need to know there is a different meaning of "shy"
"Shy" in this proverb is a verb meaning "to flinch" or "to react negatively to". We still use the word with horses (and sometimes other animals), to describe their response when they encounter scary situations and panic.
The meaning of the proverb is that if you are bitten once, you will naturally flinch away from the bite the first time - but the next time you're in a situation where you might be bitten, you'll flinch away before you're bitten. In other words, if you're in a situation where something bad happens to you, in future you'll react as if that bad thing was going to happen again.
This may be a good thing (learning from experience to avoid common hazards) or it may be a negative thing (reacting negatively when there isn't a real need to). The proverb can actually be used for both these meanings, which is interesting. Most proverbs have a "moral" meaning of what's good and bad, but this is more describing a situation and not exactly saying that the outcome is good or bad, only that it is what it is.
In answer to your question about parts of grammar, both "once" and "twice" are numeral adverbs. "Once" is the count of the number of bites, and "twice" is the count of the number of times you shy.