If you need to wash on the go, the other options are terrible: you'd need to store your nasty duds and wait for the sweet salvation of a washing machine at your destination. Or you could stop intermittently at quarter-sucking laundromats. Neither is a great option.
Here's a description of the genesis of the technique, courtesy of Steinbeck himself:
Quite early on my trip, however, I invented a method for washing clothes which you will go a long way to better. It came about this way. I had a large plastic garbage bucket with cover and bail. Since the normal movement of the truck tipped it over, I tethered it by a length of strong elastic rope of cotton-covered rubber to the clothes pole in my little closet, where it could juggle to its heart's content without spilling. After a day of this, I opened it to dispose of the stuff at a roadside garbage can and found the most thoroughly mixed and kneaded garbage I have ever seen. I suppose all great inventions spring from some such experience. The next morning, I washed the plastic bucket, put in two shirts, underwear, and socks, added hot water and detergent, and hung it by its rubber rope to the clothes pole, where it jiggled and danced crazily all day. That night I rinsed the clothes in a stream, and you've never seen clothes so clean. Inside Rocinante I strung a nylon line close to the window and hung the clothes to dry. From that time on, my clothing was washed on one day of driving and dried on the next.
That is a long way of saying, "Put the clothes in a bucket with some water and some soap." Read on for an even longer version with photographic accompaniment.