As the name suggests, it's a flying hamster wheel. Pets enter the Fly Wheel through a hole in the side and begin to run on it. The running powers the Fly Wheel's engine and takes pets to the skies.
An updated version of the Fly Wheel is mentioned towards the end of CEO Trip, but it's never shown onscreen. In the new Fly Wheel, Manny Mouser's phone app, Easy Cheesy, would be incorporated into the Fly Wheel's system as a GPS. Easy Cheesy would automatically steer the Fly Wheel to its destination, allowing the pet running on the wheel to relax during their trip.