At the beginning, your crew learns that Prof. Farnsworth has sold Planet Express to Mom, giving her control of over 50% of the Earth. For no reason at all, he's wearing a sombrero. She takes it over, makes a giant warship of it, and is planning on taking over the galaxy. Your ship is mysteriously wrecked.
After your crew flies all over the universe to try and find a way away, you finally decide to just go back in time before all this happens. Time is turned back, the ship is wrecked into the hangar, the crew steals the old (unwrecked ship), and goes to Mom's compound. However, right as you're about to run up to stop the Professor from signing it (the second time around, after escaping) the toaster (re-animator) breaks. "I guess we'll just have to be careful not to die now."
All three of you die, of course.
Farnsworth rips up the contract, yelling "Nobody kills my crew but me!" then breaks Mom's hip (she's been limping for the entire game). She offers him a sombrero, and he accepts.
The game ends with the beginning of the game.