First of all, be careful with the spelling: the phrase is "bated breath", not "baited breath". "Bait" (something used to tempt or entice, as in a baited hook) comes from an Old Norse word meaning "hunt or chase" and is unrelated.
"Bated breath", for which our first evidence currently comes from Shakespeare (but see this post about the problematic nature of Shakespearean first evidence), comes from a now otherwise defunct verb "bate" meaning "lessen in intensity", which in turn was a shortened form of "abate". "Abate", originally meaning "beat down", came from the French abattre (to beat or cut down), which in turn came from Latin.
Originally, then, bated breath was literally shallow breathing, as one would do if whispering, trying not to reveal oneself, or holding one's breath in anticipation of something momentous. The phrase therefore came to be associated with the word "wait", until "wait with bated breath" came to be an idiom in itself, meaning "be in suspense or anticipate eagerly".
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