How does the money supply expand?

Central banks control the money supply. In the U.S. the central bank is the Federal Reserve.

The Federal Reserve (Fed) has 3 main tools at its disposal to manage the money supply. They are i) the discount rate; ii) the reserve requirement; and iii) open market operations.

The discount rate is the interest rate that federal reserve banks charge to qualified private lending institutions for overnight loans accessed from the "discount window". The higher the rate, the less inclined private banks are to borrow. Thus, a higher discount rate constricts the money supply and a lower discount rate expands the money supply.

The reserve requirement is the level of funds that a depository institution must hold against specified liabilities. This relates to the whole idea of fractional reserve banking. That is, we expect that banks will make loans from their deposits but they should hold adequate reserves to meet withdrawal requests. A lower reserve requirement expands the money supply.

Finally, open market operations relate primarily to the Fed's activity buying and selling US Treasury obligations. The most active markets are in 2-, 5-, and 10-year notes. When the Fed sells bonds for US dollars, dollars are coming out of the system, thus constricting the money supply. When the Fed buys bonds on the open market with US dollars, this injects new dollars into the market, thus expanding the money supply.

There are more complicated topics related to your question that you may be interested in reading about, including: Capital Adequacy, the Velocity of Money, Fed guarantee programs.