What is the purpose of the Treasury issuing bonds? 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.