I have always thought that the US Treasury issues bonds to raise money to allow the US to pay off debts in the short- and long-term. However, recently, I have read in a book on Modern Monetary Theory that that is not the purpose of bond issuance by the Treasury. In fact, the Treasury issuance of bonds serves the purpose of aiding the Fed in conducting monetary policy. May I know if this is the case? If that is the real purpose, then what then does the Treasury do to raise money to pay off US debt?
There are some semantic issues here. A more careful statement of the MMT position is that there is no economic reason for the Treasury to issue bonds. However, there is a self-imposed institutional need to do so: the Treasury has a balance at the Federal Reserve, and the rules it follows appears to imply that the balance cannot go negative. Issuing a Treasury increases that balance, as that is where the borrowed money goes. Note that I believe that some MMT academics have argued that there is no legal need for the balance to remain non-negative - and thus this argument does not apply - but I do not have the legal expertise to judge those arguments. In any event, the Treasury acts as if this requirement exists.
However, the economic purpose of the bonds is that they offer an alternative to holding deposits at the Fed (“reserves”), and bear interest. They create a risk-free yield curve that acts as a benchmark for private sector borrowing. The Fed can set the overnight rate, and influence the rest of the yield curve. This allows it to conduct interest rate policy.
Note that although the phrasing might be disputed, the previous paragraph describes very well all standard models that include a bond market. As such, there’s no debate about the facts, rather the interpretation of the facts.
If that is the real purpose, then what then does the Treasury do to raise money to pay off US debt?
When a Treasury bill/bond expires, it is paid out of the Treasury’s balance at the Fed. This balance comes from tax receipts, and bond/bill issuance. Like corporate borrowers, the stock of Treasury bonds is continuously rolled over.
This mechanism has very little to do with the Fed’s conduct of monetary policy, it’s just the mechanics of bond issuance.
I agree with @BrianRomanchuk +1 answer which is written in the context of modern monetary theory (MMT), but an important thing to note here is that MMT is not widely accepted monetary theory in economics and in fact it has been criticized a lot by mainstream economists (See for example this paper, Krugman's blog, Cochrane's blog and so on). Since your question is more broad as you are asking if this is indeed the case let me expand on by providing mainstream thinking on government debt.
In mainstream economic thought bonds are not issued just in order to allow Fed to conduct monetary policy but for various reasons. Chief among them is that not all government debt is purchased by new created money through Fed so issuing debt is not just done to allow Fed to conduct it's monetary policy. In fact debt is a way how government can in principle raise additional funds without increasing inflation if the debt is not purchased by Fed using newly created money. This is useful to fund public projects which offer high enough returns, and it is also useful because from public economic perspective it is not optimal to always shift tax rates whenever government temporary needs more funds.
As correctly explained in Brians answer MMT argues that there is no point in Treasury to issue debt as they view it as just institutional 'theater'. This is because MMT rejects the idea that funding excessive government spending with debt will be inflationary. Consequently many MMT proponents are also arguing for expansionary fiscal policy that should always keep economy at full employment and that it should be funded through monetary expansion. If you adopt such view then you might as well argue that instead of issuing debt and buying it Fed should just 'print' (or in this day and age electronically transfer) newly created money to the treasury. This is not generally accepted idea in the profession outside of situations such as zero lower bound which are covered by standard mainstream theory.
Hence the proposition that the real purpose of issuing bonds is just to let Fed conduct monetary policy or that it is just an institutional charade is quite contentious one. In fact a more mainstream view would include raising excess resources to cover debts of the government in addition to viewing it as just vehicle for monetary policy.