In theory, there are at least two ways in which this can happen.
First, the head of the executive branch is typically an elected official, like the President in the United States. Such a person has been elected by the people (by the people of the whole country, in the case of the US) and can be said to represent their interests. When he (or some day she) directs the executive branch, we can say he is putting the will of the people into effect.
Second, it is also possible for the lower-level, unelected members of the executive branch to represent the people. The people who work for the Department of Agriculture, for example, will be in contact with farmers and will understand their needs. They will make those needs heard in meetings and will try to write them into the rules that the executive branch makes. In this way, the lower-level bureaucrats are also in touch with the people and can represent their interests.
Thus, the executive branch can represent the interests of citizens both at its highest levels and at its lower levels.