Dear Hilary King
Believe it or not, your main walking muscles are in your back.
Your spine is curved. When some of the muscles in your lower back pull on your spine it twists one side of your body forward and, at the same time, it lifts up the leg on that side. You tilt foward slightly and your leg swings forward. A step!
Of course, your leg muscles help with all this, but they're not absolutely necessary. Otherwise, people with artificial legs wouldn't be able to walk, would they?
Now the twist in your spine will make your whole body twist a little. Your legs swinging forward makes your body twist even more. Then your muscles have to work harder to make your body straight again. If you swing your arms in the opposite direction to your legs then the twisting will be counterbalanced and you will save energy. (Your arms are further away from your 'centre of balance' so, although they are lighter than your legs, they will have exactly the right counterbalancing effect.)
So humans who are designed to be able to walk all day can save a lot of energy by swinging their arms! And it feels more comfortable, too. Try walking with your arms moving in the same direction as your legs and you'll see what I mean.