3 is a common number for writing and speaking organization. I'm sure that you have had a teacher at some point assign a research paper. The rough overview is always 3 parts: introduction, body, and conclusion. More than likely that same teacher suggested that the body be made up of three main points. If you take a super basic plot structure of novels and movies, you could break those plots into three chunks. The introduction of characters and problems, the rising action/climax, and the resolution.
1984 is broken into that three "act" structure. Not only that though, the number three is all over the place in the novel. It's a bit weird at times and often makes me think of conspiracy theories. Did you notice that there are three countries? Oceania, Eastasia, and Eurasia. Goldstein's book stressed that there existed only three kinds of people due to class structuring: high, middle, and low.
"Throughout recorded time, and probably since the end of the Neolithic Age, there have been three kinds of people in the world, the High, the Middle, and the Low."
Winston works for the party, which by the way has three slogans. His job there is to rewrite history. At one point he questions why the party would do such a thing. His thoughts were triggered because of the photograph he saw of Rutherford, Jones, and Aaronson -- 3 political dissenters.
I like this quote:
"Behind Winston’s back the voice from the telescreen was still babbling away about pig-iron and the overfulfilment of the Ninth Three-Year Plan"
Crazy. A three year plan! Not to mention it is the 9th one, which is a multiple of three.
Three main characters (Julia, Winston, O'Brien).
The effect of having the entire novel broken into three parts is that it mirrors the repetition of three seen in the novel and gives readers a familiar structure by which the novel is written.