This is a phrase used in baseball when a hitter has seen many pitches but has not yet struck out. The count refers to the number of balls and strikes a batter has during an at-bat. A batter gets a walk, or base on balls, when he gets four balls, and he is called out when he gets three strikes. Going deep into the count would involve going as far as possible without hitting the limit. The maximum count a hitter can get, also known as "full count", is three balls and two strikes. If a batter is at this point, he is always considered deep in the count.
A pitcher does not want to go deep into the count; a batter does. This is because a pitcher only has so many pitches he can throw in a game, and the more he throws, the greater the chance of fatigue and mistakes. A batter wants to go deep into the count for precisely the same reasons, because they all work to his advantage. Getting deep into the count is not always as easy as watching pitches go by. A batter must pick and choose which pitches to swing at and which to let go. In doing so, a batter may foul off multiple pitches in an attempt to force the pitcher to throw a more hittable pitch. The phrase "deep in the count" comes from the amount of time it takes for a batter to get there. Getting deep into the count may be the result of any number of pitches, but it always means many pitches have been thrown.