What really happened between Hideo Kojima and Konami?

Here is the short version:

• Kojima is at the height of his powers in the 2010s. He has an exceptional franchise, Metal Gear Solid, and is an innovator in the gaming industry like few before him.
• Kojima announces that latest Metal Gear Solid game, The Phantom Pain, will be the ending installment on the main series.
• Konami announce a “corporate restructuring”. People got fired, people got hired. Kojima wasn’t mentioned anywhere. Restructure happens in March-April of 2015.
• March rolls around and Kojima gets the boot. Konami announce that they will “continue to develop and support Metal Gear products”.
• Apparently there had been internal strife between members of Konami and Kojima himself. Paired with Konami’s stick up their bum, Kojima was in a bad place.
• It is announced Kojima will be allowed to stick around until The Phantom Pain is complete though no news as to whether or not he gets to stay in the long run appears.
• Konami announces that a new Metal Gear series will be in the works, even after Kojima will possibly get the boot.
• Konami’s main argument in all of this is basically “we like Metal Gear, please don’t do this to us Kojima”.
• Another spurt of spite comes in the form of the classic subtitle, “A Hideo Kojima Game”, getting removed from the box art for the The Phantom Pain.
• Kojima leaves in October, his company’s (Kojima Productions) Los Angeles branch closes in November.
• Konami prevents Kojima from attending The Game Awards in LA in December, even though his game earns two wins. Kiefer Sutherland accepts the award on his behalf and berates Konami because Keifer is a boss.
• December 15 marked his official leave from Konami, all because the company couldn’t stand by his artistic vision.

That’s the gist. Got some info from Polygon and Kotaku - The Gamer's Guide. Hope this helps.

TL;DR Kojima has a cool game series that Konami likes a lot. Kojima wants it to end like all good things should. Konami doesn’t. Konami kicks him in the balls and steals his lunch money.