The Axiom was one of hundreds of thousands of spacecrafts buit to keep humanity safe and happy while Earth was being rejuvenated. Designed as a massive luxury cruise ship, the Axiom kept everyone aboard fed and entertained as it cruised through space. The ship was crewed almost entirely by robots, though there was a human captain. The majority of the ship's functions were handled by its computer, as well as Auto the autopilot.
Over the centuries, each new generation of humans grew more and more obese, losing bone mass in the process, due to over exposure to micro or artificial gravity and the encouraged lifestyle aboard the Axiom, thus becoming completely reliant on the ship itself. Not only that, but they became completely oblivious to the ship's original purpose, a purpose which is somewhat ironic. The Axiom was one of over 300,000 starliners (the Axiom held 600,000 and the Earth's population at the time was over 200 billion). After Buy n Large abandoned Earth and all the WALL-Es and incinerators were shut down, all of the autopilots went under Directive A113 and took full control.
Regardless, the Axiom automated computer system continued to send out probe ships in hopes that Earth would one day be habitable again. During the course of the film's events, one such ship deposits an EVE probe on Earth, and is successful in finding an example of positive plant life. It was launched in 2105, during the extensive decolonization of Earth, expecting to return five years later until Buy n Large deemed Earth to be uninhabitable and Shelby Forthright ordered all Autos to follow Directive A-113 which was to stay in space and not return to Earth.
Seven hundred years later, a probe carrying an EVE unit came back positive with a plant, proving that Earth is habitable again. According to instructions left by Shelby Forthright, the plant is to be placed in the ship's holo-detector to make an automated hyperjump back to Earth.