Quite simply, Henry evolves from an insecure highschooler into a confidant adult. Of course, baseball is intimately involved here. Why is Henry insecure at the beginning? Well, he stands in line for admission to Westish College and wonders why he is there. Henry knows that the only reason he is there is because Schwartz was scouting Henry's last game. Schwartz alone, at this point in the book, has the skinny on Henry's ability. Henry couldn't bat well, but he could play short stop like nobody's business. Schwartz ends up thinking that Henry must know where the ball is going to be hit beforehand. There is no way someone so short could move that fast otherwise. The truth is Henry is just that good. But you're right, his character needs a bit of evolution. In fact, this type of soul evolution is important, and we are told this early on:
A soul isn't something a person is born with but something that must be built, by effort and error, study and love.
Henry continues to evolve through college and becomes more confident in his own ability to play baseball. However, it's important to note an important stumbling block to Henry's evolution: through a wayward throw, he hits his roommate, Owen Dunne, while he sits in the dugout. Owen is hospitalized and Henry has a hard time recovering from this error. His skills and confidence lessen until he isn't even able to throw to first base. It takes a while for Henry to regain his confidence in his game, but when he does, he comes back full force and is eventually given a contract with the Cardinals. Note the admission of wisdom from this point in the novel:
There are no whys in a person's life, and very few hows. In the end, in search of useful wisdom, you could only come back to the most hackneyed concepts, like kindness, forbearance, infinite patience. Solomon and Lincoln: This too shall pass. Damn right it will. Or Chekhov: Nothing passes. Equally true.
The full evolution of Henry's character ends in the final scene of the book where Mike and Henry are simply honing their baseball skills while they have fun with each other, using a shovel to mark a "perfect throw." Henry achieves that perfect throw before all the balls are used. Henry's smile shows his confidence achieved.