Why do you think Orwell organized 1984 into 3 parts? What effect does it have on the reader?

It is quite common for plays to have a three act structure, and Orwell applies it to 1984. The first act normally encompasses the exposition; the second act, the rising action; and the third act, the climax and resolution. Adopting this structure feels particularly appropriate as it allows us to accompany Winston Smith on his long and tortuous journey, not quite knowing where it will lead, but certain nonetheless that we can expect a suitably dramatic climax.
Appropriately enough, this journey passes through three distinct stages: Winston as humble state functionary; Winston as rebel against the regime; and Winston as the physically and mentally destroyed victim of torture who's learned to love Big Brother.
It is also notable that the number three comes to have a great deal of significance in the book itself, cropping up in numerous contexts. There are three world powers (Oceania, Eastasia, and Eurasia), three main characters (Winston, Julia, and O'Brien) and three main slogans ("War is Peace," "Freedom is Slavery," "Ignorance is Strength.")