This statement is not literally true. The FOMC is not officially charged with doing anything with the financial markets. However, it is certainly true that the FOMC's actions can affect financial markets.
The actual role of the FOMC is to help regulate the supply of money in the American economy. The FOMC is responsible for open market operations. This means that it is in charge of deciding when to buy or sell government securities so as to affect interest rates in the US. The FOMC typically sets an interest rate that it wants as a target. It then buys and sells government securities so as to achieve that target rate.
When the FOMC does this, it is trying to maintain stability in the US economy as a whole and, as a result, in the financial markets. It is trying to ensure that the US economy grows without experiencing inflation.
The charge of the FOMC is much greater than simply making sure our financial markets are stable. Instead, it is supposed to try to make sure that our economy as a whole is stable by keeping the interest rates at a given level that it deems most compatible with stable economic growth.