Ok let's see. A lot of right and wrong.

Bungie had sold off Halo to Microsoft a long time ago. Bungie wanted to make games for more than 1 console, but Microsoft wanted to keep Halo as an xbox exclusive.

So when Bungie's contract ended, they left Microsoft to create their game for multiple consoles. Bungie eventually teamed up with Activision to do this.

It is not certain that Bungie wanted to leave the Halo Franchise or not. They just didn't want their skills to be held back to one console. They wanted to develop it for more people to experience.

Bungie didn't leave Halo, they left Microsoft, but Microsoft owns the rights to Halo so Halo didn't go with Bungie.

343 was not made after Bungie left. 343 has always been around. Bungie was the big name over Halo, but 343 has always been around to help in the shadows.

When Bungie became independent, many members of Bungie had grown attached to the Halo Universe. Some members of Bungie switched over to 343 to continue the story of Halo. 343 has members that have worked on Halo before Halo 4. (Bungie members that switched)

343 is not new to Halo. The community is just new to 343.

Bungie didn't wanna be ruled by Microsofts crap any more.