Charlie: Did you watch the political debate last week?
Emilio: Of course! It was so shocking; both candidates came out swinging.
Charlie: That’s an understatement; they were beyond aggressive.
Emilio: They attacked each other relentlessly.
Meaning: To "come out swinging" comes from the aggressive act of swinging your arms to punch someone or an object to hurt others. It’s used as an expression to depict someone mounting a verbal attack on someone aggressively. In this example, Emilio describes the politicians having a debate which probably involved strong accusations, strong/aggressive behavior or language to attack each other.
Sarah: If Peter doesn’t have a good explanation for leaving me and my family in the middle of dinner, I’m going to come out swinging at him when he arrives.
Michael: Hey, take it easy. Don’t be so violent. I’m sure there’s a rational explanation. However, that doesn’t give you the right to attack someone physically.
Sarah: I won’t actually do it, I’m just so angry for his rude behavior.
Michael: I understand your anger, but saying something like that will confuse people and think you’re violent.