The usual modern practice is for central banks to set a policy rate, which is where it wants short-term (often overnight) interbank lending rates to be. This then becomes the reference rate for other interest rates. An interest rate is just the cost of borrowing (expressed as a percentage per year); it is itself not money.
The rate of interest is set to achieve policy objectives, such as controlling inflation. (I will not attempt to answer the part of the question whether raising the rate of interest itself causes inflation, as that us extremely controversial.)
There is no need for the central bank to actually pay that rate of interest on anything, as I will discuss later.
It is possible for private banks to borrow at that interest rate from the central bank. In the United States, that would be discount window borrowing. Banks would do this if they need funding and cannot get it elsewhere (it is normally frowned upon). The discount rate is another policy rate that is set as a spread to the main policy rate (the fed funds rate). However, since the private bank is borrowing from the central bank, money is flowing from the private sector back to the central bank, reducing the money supply.
Pre-2008, the Federal Reserve did not pay interest on balances held at it by banks (“reserves”). That is, it could keep interest rates at a target level without paying interest to do so.
However, it now pays interest on reserves. It needs to do so now in order to keep inter-bank rates at its target rate as a result of the decision to create large amounts of excess reserves (“quantitative easing”). The payment of interest by the central bank creates “money,” since balances at the central bank are part of the money supply. (This might be termed “money printing,” but that phrase is somewhat vaguely defined.) The increased balances are created by the central bank adjusting upwards private bank balances (in the same way private banks do to their customers), so the money “appears out of nowhere.”
Does this payment increase the debt of the country? Not by standard definitions, since they usually equate governmental debt to the amount of bonds/bills issued by the fiscal authority (the Treasury in the United States). However, it increases the money issued by the central bank, and money is a liability of the central bank. Since the central government owns the central bank, this means that governmental liabilities are higher.