First of all, the energy savings. After the refrigerator, the dryer uses the most energy of all your appliances. The only reason the refrigerator is the energy hog winner is that, unlike the dryer, it is on all the time. Second, the clothes wear out more slowly. Basically your dryer is beating your clothes to smithereens every time you use it. That is what lint is- clothes smithereens.
Third, less ironing. When clothes are hung up on the line they are flat, when the are in a dryer they are wadded up in a little ball. It stands to reason they would need more ironing when they come out of the dryer.
The clothes also have a unique, fresh smell. I have no idea why, but especially for your sheets, it is really pleasant.
And finally, it makes me slow down and spend a few minutes outside in my garden. Like many moms, and small business owners, I have a hard time taking a moment during the day to really be aware of the world around me. Whole days can pass without my thinking of one thing beyond my 'to do" list. When I take the time to hang up a load of laundry I have to slow down for 10 minutes. I have to stand in the sun, on my lawn, and feel the sun on my back. My mind can wander, because pegging clothes on the line is a simple task, and yet I am occupied enough that I stay in the moment.
You know, I think this last might be the most valuable benefit of drying my clothes outside.