The Fed more or less controls interest rates using open market operations, but why?
Why are interests rate not controlled by supply and demand?
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The fed controls interest rates via supply and demand.
The fed directly controls only one rate, the rate at which banks can borrow from the government.
Lowering this rate permits banks to lower the rates they charge while making the same profit. They don't have to do so, but supply and demand says that those banks who do lower their rate can take business away from those which don't, so the market pressures them to more-or-less follow the fed rate's shifts.
The Fed doesn't control interest literally. They attempt to influence the rate banks charge each other for short term loans using the "open market" to inject and take away reserves.
Why does the Fed overshadow this particular segment of the economy (which granted correlates somewhat with other interest rates)?
It's all about propping up the banking system. Inflation/growth...they will claim these goals are important, but in practicality the Fed's primary worry is making sure the banking system doesn't melt down. It's a legit worry. Banks operate by mismatching short term low yield debt with long term high yeild assets. An inherently insolvent situation in the aggregate that requires constant indirect bailing out from the Fed.
The Fed has determined that targetting the Fed Funds Rate is the best way to do this. It didn't use to be this way. They used to target the discount window early in the 20th century. Out of fear that primary dealers were churning the open market (they are), they tried targetting M1 (should have been M3) during the Volker years in the early 80's. But ultimately the Fed has preferred Fed Funds targeting as it seems to reduce volatility in the banking system (but increases aggregate risks and inflation).