This is a baseball term which refers to movement, usually a hit, along either the first base or third base lines of the field. These two lines serve as the foul lines, or out-of-bounds lines, on a baseball diamond. They both start at home plate and proceed outward, 90 degrees from each other, through first or third base, and all the way back to the outfield wall.
A hit is considered to be moving down the line if it moves along either the first or third base line, in any speed or fashion. The ball can be bouncing, in the air, or rolling along the grass; it is irrelevant in this context. If a ball is hit down the line, it can be very difficult for the defense to get to because the first and third basemen do not always play up against the edge of the field. It is rare for a player to hit the ball down the line.
In addition to referring to a hit (its most common usage), "down the line" can also refer to a player running. An offensive player may be running down the line if he is trying to reach base, while a defensive player may be running down the line to chase down a ball or an opposing player. As long as the ball or the player is moving along the line, and following a relatively straight path, this phrase applies.