The home was invented by Hector in order to escape the Village of Fowl Devotees. It was just like a large Hot Air Balloon, but instead of one basket, there were twelve, tied together below several hot air balloons. Each basket was large enough to serve as a different room, making it just like a flying house. It was self-sustaining, so it was not only unnecessary to return to the ground, but impossible. The engine was designed to last for a hundred years, and a storage basket was filled with food, beverages, clothing and books.
Although one would question why he couldn't simply leave normally (as it seems legal for village members to leave), it is implied that Hector built it so that he could escape the troubles of all the world for the rest of his life. He could also have been trying to escape VFD; he says he wants to leave "V.F.D. and the Council of Elders", and he is known to have been a member of VFD before taking up residence in the Village of Fowl Devotees.
When the Baudelaires began to follow and climb up the long rope ladder, Olaf and Esmé Squalor appeared. Esmé shot the rope ladder with a harpoon gun, causing it to unravel. She also caused further damage to the vehicle by harpooning it.
The Baudelaires decided they had to climb back down since if they kept trying to climb up, they would fall to their deaths. Violet was especially devastated and shed a tear, as she was uncertain if she would reunite with Hector and the Quagmires again, or if they would be safe.
The balloon home disappeared into the horizon, forcing the Baudelaires to flee to the Hinterlands in order to escape the mob of angry villagers demanding them to be burned at the stake.
Duncan, Isadora and Hector somehow sent a message that made it seem like they were in trouble, as they were being attacked by V.F.D. eagles. Kit Snicket and Quigley Quagmire received the message and stole a helicopter from a nearby botanist. Only Quigley went to save them, as Kit had to help the Baudelaires.
Quigley managed to reach the self-sustaining hot air mobile home and helped his siblings and Hector catch the treacherous eagles in an enormous net. However, the home was later destroyed when it crashed into the Queequeg after suffering damage from the eagles.
TV Series Divergent Canon
The home does not crash into the sea in the Netflix canon; it is unknown how Hector, Duncan and Isadora escaped the eagles, but they are seen picking up Quigley Quagmire above Anwhistle Aquatics, and the home can be seen floating outside a window while Lemony Snicket narrates, approximately ten years after the events of the series.
It should also be noted that Hector does not seem to be affiliated with VFD in this canon, and thus the ship was likely built only to escape the Village of Fowl Devotees.