Why did Gatsby buy a house across the bay from Daisy?
East Egg, where Daisy and Tom Buchanan live, is for the super-wealthy who come from “old money.” West Egg, where Gatsby buys his mansion, is for the recently wealthy, or “new money.” Gatsby has only recently become wealthy through some questionable dealings with Wolfsheim, and Gatsby must proceed with caution in trying to win Daisy’s love. Daisy is married and seemingly above Gatsby in social stature. Gatsby begins to woo her by throwing lavish parties and hiding who he is until he finds an ally in Nick who will introduce them. Gatsby must prove he is worthy of Daisy’s love first because he doesn’t really know what her reaction to him will be. Therefore, he proceeds with some apprehension in approaching her. Moving next door would have been too bold a move and would have shown Gatsby as too aggressive. So, for a while, Gatsby tries to find ways to prove his worthiness and contemplates his future life with Daisy as he watches the mesmerizing green light blink on the other side of the lake from his boat dock in West Egg.
Gatsby has purchased this house because of its proximity to Daisy. He has loved her, or at least the idea of her, since they met. I think that since he has loved her from afar for so long, he needs to be close to her, but not too close. He is comfortable gazing at her from a distance before they meet again, and even after they meet, there is something romantic, idealistic, and dreamy about his love that makes him want to view her from that distance. Then he can fantasize without being confronted with the reality of her (or her dreadful husband.)
Of course, as a practical matter, he wants to meet her again, and what better way than to buy a fancy house in her "neighborhood", have large parties, and hope she turns up? Once he meets Nick, he finds a better way of accomplishing his mission.