When watching an intense football match, you might find yourself waiting with bated breath, or in intense anticipation and nervousness, for your beloved team's next move.
Abated is the past tense of the verb abate, "to reduce in intensity," and bated is a shortened form of abated. So with bated breath means "with one's breath reduced" — as someone might hold their breath when excited or anxious about something about to happen. The phrase was popularized in Shakespeare's play The Merchant of Venice, in which it described (mockingly) whispering subserviently. Today, however, it's used to capture the sensation of anxious, excited anticipation.