Under the Constitution of the United States, only the legislature (Congress) can authorize money to be spent. That is one of Congress’s most important functions: its power over the federal budget. That money, however, is allocated to departments and agencies that fall under the purview of the Office of the President. The Departments of the Treasury, of Interior, of State, of Defense, of Health and Human Services, and so on, are all headed by individuals selected by the president, with the advice and consent of the Senate. The day-to-day operations of these departments are all a function of the powers of the presidency. That is why they are called “Executive Branch agencies.” These agencies, however, cannot function without money, and that money can only be provided by the Legislative Branch of the government. Under Article I, Section 9 of the Constitution, Congress is given explicit authority to control the money (“No Money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law . . .”). Bills that specify how much money each Executive Branch agency receives must originate with the legislature, and those bills regularly go into minute detail specifying precisely how, when, and where that money will be spent. That is a lot of power, and is integral to the system of checks and balances established in the Constitution.
With the "power of the purse strings" so explicitly given to Congress under the Constitution, how can the president increase or decrease the scope of the government? By not spending the money allocated by Congress, for one. Throw a rock in the nation’s capital, and there is a good chance you will hit a lawyer, and a smart one at that. Many of these lawyers specialize in manipulating processes for the benefit of their clients, and those clients can be government contractors, foreign governments, special interest groups of all manner, etc. Laws can be and are interpreted in many ways, often despite the existence of documentation specifying the intent of those laws. Presidents have innumerable sources of wisdom regarding legal interpretations, and those interpretations are often designed to subvert or undermine Congressional intent. It is not uncommon for lawyers from a federal agency to argue the meaning of laws with the same elected officials who wrote those laws in the first place.
In short, presidents can increase or decrease the scope of government by ignoring congressional intent, often by refusing to spend money appropriated by Congress for a specific purpose, and they can influence the scope of government by the zeal with which they do or do not comply with congressional mandates. They can refuse to spend money to hire more government workers, or they can refuse to spend money intended by Congress to establish a new agency, or they can establish the agency but refuse to adequately staff it so that it can perform its mission. These tug-of-wars between the Legislative and Executive Branches of government take place every day, and they illustrate ways in which presidents can influence the scope of government.