Why are bond interest rates set at the interest rate set by the Fed?

Bonds are offered at different interest rates; the market works to make them effectively equal.

Imagine the Feds are offering 1% interest. Now someone else (X) offers 1.1%. Why in the world would you (or anyone) buy Fed bonds that give 1% when you can have 1.1% by buying from X (assuming X is equally stable and secure as the Feds)?
What happens is that people try to sell the 1% bonds and instead try to buy the 1.1% bonds. When a lot of 1% bonds are on the market, and nobody wants them, their price falls, so you can buy a 100$ bond for 99$, 98$, etc. At the same time, the 1.1% bonds have lots of people that want to buy them, and not many sellers, so the price goes up - a 100$-bond goes for 101$, 102$, etc.
This continues until the efficient interest rate of both is equal again: if you buy the 1% bond for less than its nominal 100$ value, you can still return it after its lifetime for 100$, so you make effectively more than 1%. Same for the 1.1% for over 100+x.

In a nutshell, the logical behavior of the people in the market equalizes the effective interest rate of equally secure bonds.

I have a question about bonds: why do corporations, municipalities, and the treasury set bond interest rates at the current U.S Federal Reserve interest rate?

Simply they don't set the rate the same. The risk of defaulting on a bond is different depending on who issued the bond. There are stories about companies who are having trouble paying their bonds even when the economy is good. Those companies struggling to pay may be heading for bankruptcy. The next time they need to sell bonds they will find they have to promise a higher rate to the purchasers.

This risk of default is even true with governments. Cities, counties, and states worry about how much they will have to pay. Growing tax income and controlling spending saves the government money when they need to issue new bonds. And yes even governments with growing tax revenue and controlled spending issue bonds. But sometimes if the government is struggling financially they have to pay a higher rate of interest.

There are companies that rate the bonds of governments and corporations. The ratings of these bonds impact the rates the issuer has to pay with their next set of bonds. Companies and governments worry about their rating because a poor or junk rating will make borrowing more expensive.

meta: not really an answer because the question's premise is wrong, but no longer fits in comments

What goes into a bond's yield (and price)?

According to economic theory, people are rational and investors in particular will choose investments that maximize their likely wealth. Since all investments have some downsides, more or less likely, but not the same ones -- risk of not gaining, or gaining less, or losing some or all, risk of missing out on something else -- and people (even investors) have somewhat different responses to (different) harms or risks, this is usually described as maximizing 'risk-adjusted return' across the range of possible investments.

Thus for a given bond, the return demanded by 'the market' can be broken into two parts:

• the 'cost' of being invested for a given duration in any bond, because having $x now is almost always preferable to (and thus more 'valuable' than) having 'equal' $x at a future time which might be anywhere from days or months to decades depending on the bond. In addition, generally prices (the relation between money and things) rise over time, called inflation; this is usually gradual, and in the period since the financial crisis of 2008 has been espeically subdued at least in what we call the 'developed' world. Generally all investments must expect/promise to return future $y with $y > $x to compensate the investor for the 'time value of money' plus inflation. But how much this premium (or equivalently discount of the future amount(s)) must be varies depending partly on people's expectations of the future (particularly for inflation) and the availability of (i.e. competition among) money to be invested. This is the part the Fed 'controls' -- or at least strongly influences -- by adjusting the 'money supply'.

• the additional risk, and possible cost, of investing in a particular bond compared to others. Most of this is due to risk of default by the issuer -- i.e not paying the required interest (at all, or on time) and especially not repaying the principal (ditto) -- as detailed in mhoran_psprep's answer. More exactly, since we can't know the actual risk of things happening in the future, in some cases far in the future, it is perceived or estimated default risk that affects investors decisions on what returns to demand in compensation for this risk. In addition to the risk of issuer default, there can be other risks:

  for an issuer in a foreign country, or more exactly using a foreign currency, there is the 'currency risk' that $x converted to #y today may not convert back to $x in the future

  there is 'political risk' or 'country risk' that a government may prohibit, restrict, delay, tax, or otherwise disadvantage repayments. This can in principle happen anywhere, but historically it is more commonly applied to foreigners than locals (who are much more often voters, or at least constituents).

  for a long-term bond there is not so much a new type of risk, but additional time in which other risks may manifest, and can change over time in ways that are difficult to predict (and thus value) now

Thus the return on a particular bond will go up or down, and its price move inversely, depending on both the 'current' rate for all bonds of that duration, i.e. the yield curve which may move daily, and the (estimated) risks of that particular bond, which typically move only when the organization's operations (are perceived to) change e.g. they report quarterly results or a merger or bankruptcy or such, or the economic and/or political environment changes, which does happen but not every day.

This is why the Fed is criticized in the current pandemic (and other central banks too) for not 'focussing on' parts of our society and economy that are in (even) greater distress than average. When the Fed eases or tightens, it does so for everybody equally; if the market rated company A a better investment than company B at a Fed rate of 3%, and the Fed goes to 0% with no other change, the market will still prefer A and B may be left in the cold. (Or given global climate change, the hot and flooded :-)