How Does Praying Make You a Better Person?

"The Holy One blessed is He desires strongly the prayer of the righteous" (Yevamos 64A)

The reason for this "strong desire" is because He strongly desires their perfection, and Prayer is indeed a means of gaining great perfection of mind and character. In order to reward His righteous ones, Hashem sometimes even exerts considerable pressure upon them (such as rendering the childless for a long time, as in the instances of Abraham and Sarah, Isaac and Rivkah, Rachel and others - ibid.) in order to elicit more prayer from His chose ones, so that the greatest achievements of mind and soul be attained by them. The grapes that yield the best wine are pressed to extract every drop; and it is this greatness, derived from prayer, that Hashem desires strongly for them. -- Praise, My Soul