Constitutional FAQ Answer #107
Q107. "Where does the C.I.A. get their power and authority?"
A. The CIA, and a whole host of other government agencies, fall under the Executive Branch. Their power, then, comes from the power of the President. The job of the executive branch is to carry out the legal authority given to it by the Legislative Branch, the Congress. So the power and authority of the CIA comes from the legislation authorizing the creation and funding of the CIA, from the powers delegated to it by the President and/or the Congress, and from the prestige and respect of the agency by other parts of the government.
The same applies to every executive department or agency. From the FAA's power to regulate the skies over the United States to HUD's ability to regulate housing standards, the power comes from the people in the form of the Congress. Congress can curtail the acts of an agency at any time with appropriate legislation (though once an agency is established, it can be hard to disband; and once a power is given, it can be hard to take it away).
The question also begs another: why do we even need agencies like the EPA, the FBI, or the FDA? The following quote from Constitutional Law: Cases and Commentary by Daniel Hall says it quite well:
... the job of governing has become too large for Congress, the courts, and the President to handle... Congress does not have time to make all the laws, the President to enforce all the laws, or the courts to adjudicate all the cases... Congress is too small to be an expert in every subject. Agencies, however, specialize and, as a result, they possess technical knowledge and expertise in their subject areas."