Knowyourphrase.com explains the origin of this phrase is likely baseball and that the first published examples of its use were found in the 1880’s. I seldom disagree with sources, but the first explanation is that the batter makes a quick decision to run to first base after hitting the baseball. I have the opinion that it refers to a fielder moving immediately after the bat hits the ball to a place to make a catch. Regardless, the expression is used to do something immediately, in a hurry, or without delay.  The example sentence involves warning a person considering buying a home who is told “right off the bat” the house is infested with termites!