How and why was the Battle of Bougainville fought?

Here's a short, partial answer.

The Japanese stronghold at Rabaul (New Britain) was within range of American bombers based on Guadacanal, but their fighter escorts could not accompany them the entire distance. This was, of course, well before in-flight refueling.

If the Americans could place an airfield suitable for fighter planes on Bougainville Island, they could base the fighter escorts there, and protect bombers heading for Rabaul.

The American forces landed at a coastal region of W Central Bougainville, called Torokina. They only intended to control an area large enough for the airfield; there was no intent to mount a land campaign against the very large Japanese force on the island. Of course there was fighting but before long the field was secured and operations began.

An excellent summary can be found in: Gailey, Harry A. Bougainville, 1943-1945, The Forgotten Campaign. Univ. Press of Kentucky, 1991.

There are, of course, many other printed resources, and this is only a short, partial answer to the question.