Why does the legislative branch of the government represent the people most directly? what is the principle job of this branch?

The principle job of the legislative branch is to make laws, and those laws should be in accordance with the will of the people - that's why the legislative branch represents the people.

The juidicial branch represents the people the least, since it checks the correctness of the laws and conducts court cases.

The executive branch can be argued to work semi-independently of the will of the people, executing the laws. But the laws need to be passed in accordance with the will of the people, so the legislative branch must represent what the people want.