Is this one thing credited with saving the airline, or were there other efforts involved? I don't understand how a 10-minute turnaround time can turn an airline around so quickly.
Well it's pretty straightforward, if turnaround time is only 10 minutes and your competitors take an hour then you can simply make more flights per aircraft per day than them. This is amplified when your flights are short distance domestic. Depending on destination it probably allowed them to make 2 - 3 extra flights per aircraft per day, at a time when jet fuel was not so expensive. That could make a massive difference to profitability.
Hell, even in a day where jet fuel is very expensive, daily miles is still a key factor. Your margins might be smaller thanks to jet fuel, but more flights still means more revenue.
There were other efforts involved. Actually, if you're curious, there's some good analysis of Southwest Airlines' success in a book I read recently: "From Good To Great" by Jim Collins
My apologies, the book name was Great by Choice, as someone else pointed out. Both books are excellent and worth the read though.
It seemed that it made up for the one airplane they had to sell, which destroyed their planned schedules, but it wasn't super clear. Maybe if they hadn't met previous obligations they would have gone under.