An economic approach to the issue would be to start by identifying the conditions under which hedging would be irrelevant. If trading is costless and markets are completely efficient, then hedging wouldn't add value. Under these assumptions, any transaction that the firm does can be undone by the shareholders outside of the firm at the same relative prices. So we look for market imperfections of one sort or another to explain hedging.
In most cases, the market imperfection is the cost associated with financial distress or bankruptcy. If unexpectedly high operating costs need to be covered by borrowing, and if borrowing is costly when done on short notice, then it makes sense to smooth out the variation in operating costs. Hedging--in this case, locking in a forward price of a key input to production--allows that to happen. This theory cannot explain why Southwest hedges its fuel costs and the other airlines don't, because it is in the best financial shape.
Perhaps we can tweak it a bit (as was being done in the comments) to suggest that Southwest is one of the few airlines mentioned where the stockholders are actually the residual claimants. The other companies are much closer to bankruptcy, when the equity holders get essentially nothing and any assets get assigned to the debtholders. Refusing to hedge in this case is a form of risk-shifting onto a financially weak firm's creditors. It may also be that the weak financial position of the other airlines doesn't allow them to enter into the long-term contracts involved with hedging next year's fuel costs.
Another possibility is that Southwest has a very unusual business model and is a $10 billion company because it rigorously applies that business model and looks for ways to improve it. If they lock in the price of their fuel, then fluctuations in the price of fuel won't interfere with their ability to figure out what routes are profitable, what schedules improve efficiency, or what the next city on their route map should be. Variation in performance month-to-month will better reflect choices they made rather than fluctuations they couldn't control.
But maybe this is overthinking the problem. Two months ago, David Grossman wrote in USA Today:
Southwest reported a profit of $235 million and saved approximately $351 million during the first six months of this year. If Southwest hadn't hedged, that profit would have been a $116 million loss and the first time in 57 consecutive quarters that the company did not report a profit.
Maybe they hedge to keep the streak alive.
This topic comes up any time fuel costs increase. About a year ago, Jim Garven noted most of these points and provided links to empirical and case studies of fuel hedging. Some other good discussions are here and here at the Conglomerate blog.