There’s two somewhat related points that give a simplified answer to your question.
- The government (central bank) sets the base interest rate in a free-floating currency.
- The government will set that interest rate in a manner that meets its policy objectives; the private sector has no incentive to meet those objectives on its own.
I will outline these in turn.
If we are discussing a free-floating currency (e.g., no gold conversion right), the central government is the monopoly supplier of the monetary base. Roughly speaking, all private monetary claims can be viewed as a claim on that government-issued money. As the monopoly supplier, the government can dictate the price of money - i.e., the interest rate.
More specifically, private sector debt arrangements are typically set up with liquidity backstops. For example, commercial paper lines are backed with credit lines at banks. Some banks will go to other banks for backstops. However, those banks need a backstop - the central bank. In a liquidity crisis, the central bank sets the interest rate for emergency lending to banks. The interest rate for such lending becomes the base rate of interest in the economy. This rate is determined solely at the discretion of the central bankers, and thus is under control of an arm of the government.
(Hyman Minsky discussed the liquidity structure of banking systems in detail; my description here is my own phrasing. The book “Stabilizing an Unstable Economy” discusses the evolution of the financial system of the United States in the post-war era.)
Once we accept that the government sets the base rate of interest, how does it set it? That leads us to the whole theory of monetary economics, which I cannot hope to summarise. The standard story is that the central bank sets the interest rate to control inflation.
Why won’t the banks do this? Private sector lenders only care about getting paid back. If the economy is growing rapidly in nominal terms, it means that borrowers’ incomes should be growing - and they can easily meet debt repayments. So there is no reason to fear inflation.
One could argue that bankers are worried about the real return on lending. However, this ignores the nature of banks. Banks are highly levered entities, and so their return on equity (what they care about) is much higher than the level of interest rates - so long as borrowers do not default. Therefore, they have no reason to care about the inflationary consequences of their lending decisions.