How it all began...
The post-apocalyptic Land of Ooo's condition was caused by the Great Mushroom War.
The inclusion of the word "Mushroom" refers to the war's nuclear nature. The creatures and characters in the Land of Ooo are then supposedly mutations due to the radiation of the nuclear bomb dropped on Earth. It can be inferred that the nuclear Mushroom Bomb brought about the end of the war and birthed and created the Lich (the ultimate representation of evil throughout the show) around 1000 years before present day. It is believed that only Marceline, Abraham Lincoln, Moe (who is now a cyborg) and the Ice King were the only ones to survive the Mushroom War.
There's a lot of proof of war and prior inhabitants if you pay attention to the surroundings and details in various scenes and episodes. Check it out:
Two dead people sitting on a couch in their post-apocalyptic home. Cars, trucks, traffic lights and debris is spread around their home
The opening scene of Adventure Time's theme song depicts the world after the Great Mushroom War. Missiles from the war, random broken belongings and a polluted grey sky with dead trees is a result of the nuclear war.
A drawing of the Earth being attacked drawn by the Ice King hidden in his belongings
The Mushroom Bomb explosion and the birth of the Lich
Excuse the spelling errors... it's a more visual representation of the Great Mushroom War.