Ray Kinsella's birthstone is a diamond. This is fitting because baseball is played on a diamond-shaped field. Ray is a baseball fanatic. When he asks himself why he would want to carry out the demands of the "voice" telling him to build the field, he says, "But as I ask, I already know the answer. I count the loves in my life: Annie, Karin, Iowa, and Baseball. The great god Baseball." (This is early in Chapter 1.)
To Ray, baseball is like a God. He loves the history of it, the mythology of it to the extent that it is like a religion to him. While watching the Red Sox game, he says baseball "is the most perfect of games, solid, true, pure and precious as diamonds." Again, there is diamond symbolism relating to the field in that, like baseball, diamonds are precious, incredibly hard (and therefore unchanging, pure and natural). So it is no surprise that Ray makes the joke (although partly serious) that his astrological sign is "hit and run." He also notes this early in Chapter 1.
A "hit and run" is a play in baseball in which a man on base steals the next base at the same time the hitter swings to hit the ball. The idea is that if the batter hits safely or on the ground, the lead runner gets a head start. So, he uses this phrase simply because it refers to baseball and he is a baseball fanatic.
But why "hit and run?" Why not "double play" or "grand slam"? There doesn't seem to be a clear answer to this, so it is up to your interpretation. It could have something to do with the themes of the play relating to relationships, fathers and sons, and so on. Or, since the "hit and run" succeeds only when things fall into place, this might have some relevance to the magic of the field. A "hit and run" is all based on timing: stealing at the right time, hitting to the right place. At the end of Chapter 2, Ray talks about the magic of the field:
It's the place and the time. The right place and right time. Iowa is the right place, and the time is right, too--a time when all the cosmic tumblers have clicked into place and the universe opens up for a few seconds, or hours, and shows you what is possible.
Things happen at the right place and time, like a successful hit and run. Going by this interpretation, Ray might feel lucky that he was born at the right place and time, stressing emphasis on his astrological sign, which relates to the time he was born.