It means "immediately," "right way," "first thing." I don't know why it means that, it is a true idiom. For example, consider these two conversations. Conversation #1: Rob: "Hi, Dan." Dan: "Hi, how are you?" Rob: "I'm fine, how are you?" Dan: "I'm great. How's your family?" Rob: "Doing great. Say, Dan..." Dan: "Yes..." Rob: "I was wondering, could you do me a favor?" Dan: "Sure, what?" Rob: "Our lawnmower is broken, could you lend me yours?" Conversation #2: Rob: "Say, Dan, could you lend me your lawnmower?" In conversation #2, Rob asked for the lawnmower right off the bat. To introduce a few more idioms: in conversation #2, he didn't beat around the bush. He got right to the point.

It's like when my wife comes home from work and starts to complain before even saying hello. She complains right off the bat.

To start something “early” or quickly

The others are correct. I assume it comes from baseball or cricket . . . i.e. when the ball is right off the bat, i.e. just after being hit.