It is about baseball. Carl Yastremski was a famous baseball player, now retired, who played for the Boston Red Sox of the American League in baseball.
If "Yastremski is up with a man on" it means that Yastremski is batting and there is one man on base.
There are four bases, with the batter being at one of them. By hitting the ball where the other team can't catch it, the runners can run to the next base, possibly touching all four bases, which would score a run (point).
If you want all the rules (and there are many) search for 'baseball' on the internet.