While it is well documented how Hideo Kojima and Konami parted ways, many of the facts and events are just rumors and not official details or facts. It is highly probable that there is a NDA (non-disclosure agreement) that Konami forced Kojima to agree, the well-known gaming creator speak about the genesis of the new Kojima Productions.

Sony and Hideo Kojima are co-producing a documentary that focuses on Kojima right after he was ousted and officially finished his tenure with Konami.

The documentary Hideo Kojima: Connecting Worlds is directed by Glen Milner and its billed as providing "a rare insight into Hideo Kojima's creative process " during the launch of his new studio and features interviews with several of the developers collaborators, including Guillermo del Toro, Nicholas Winding Refn, Grimes, George Miller, Norman Reedus, Woodkid, Chvrches and more. 

The documentary is a joint production between Kojima Productions, Filmworks, and PlayStation Studios and slated to be released soon.

What do we know or heard about the Konami and Kojima saga?

While details has not been clear (and probably will never be), Kojima's career on Konami ended because of a constant power struggle within the company as newly appointed executives wanted Konami to focus on mobile gaming/content and saw Hideo Kojima as VP as a main obstacle because his lack of interest on going away from AAA and his habitual long time of producing a game rubbed people against him the wrong way.

Reportedly he let his detractors a golden opportunity against him after Metal Gear Solid V: The Phantom Pain apparently, failed to be within development budget as Konami conditioned to green light another Metal Gear Solid game and was used to justify to remove him from duty and delegated a only a contractor with limited access to Konami assets.