What are the functions of the base and the arm of microscope?

The arm holds the tube, containing the optics, along with the eye piece and the focusing mechanisms. It is designed for great strength so, it can also be used as a handle while carrying it.

The arm supports the tube and connects it to the base. The base keeps the microscope steady and prevents any unwanted wobble or shaking. It also acts as a support for the entire scope as well, to stabilize and protect the microscope.