Airlines have a large fixed cost and a small marginal cost for each passenger on a plane. If they charge $250 for each seat and half the seats are empty then they are wasting capacity and have to charge enough for each seat such that half the seats pay for an entire flight. If they can fill the second half of the plane with $50 passengers then they are providing more service (a social good!) and can perhaps reduce the expensive tickets to $200.

Of course everyone wants the $50 tickets, so the market gets all weird trying to both fill flights and charge as much as possible, while being competitive with other airlines that are trying to do the same thing. Because there are competing airlines, the final margins stay low, so the airlines are barely "maximizing" profits so much as they are simply maintaining profitability.