What an empty leg flight actually is
An empty leg is a one-way private jet flight that an operator already has to operate, with no booked passengers on board. They happen for two reasons: a paying client booked a one-way charter (say, Austin to Aspen) and the aircraft has to return to its home base, or the aircraft has to reposition to pick up its next client on the opposite end of the route. Either way, the aircraft is going. Selling the seats — or the whole cabin — at a steep discount turns a cost into revenue.
This is not a different kind of aircraft, a different operator class, or a "budget" version of charter. It is the same Part 135 lift, with the same safety standards, at a fraction of the price.