Why do leaves of trees drop on the east side of us in fall?

Actually the leaves on some trees drop everywhere in fall (not just in the eastern US) and the leaves on other trees never drop.

The trees that drop their leaves in fall are called deciduous and the trees that never drop their leaves are called evergreen.

The reason for the phenomenon you are describing is that in the eastern US forests have many more deciduous trees than evergreen trees, while in the western US forests have many more evergreen trees than deciduous trees. But forests on both sides of the US always have some of both types.

Deciduous trees drop their leaves in fall mostly as a way to minimize water loss in winter to the cold dry air. Evergreens have an entirely different means of coping with cold dry air in winter.