In essence the scheme was designed to solve two problems at the same time, the fragmentation and uncertainty of the Revolutionary Debt and the chronic shortage of cash, gold and silver, circulating in the US economy.

While rich in land, crops, timber and other assets, the United States also had great demand for imported goods and few liquid assets. Whatever gold and silver coins were present in the US tended to leave it quickly to buy imported goods, which were used to expand the production of agriculture (i.e. more land, crops and timber) and such and there were few if any gold and silver mines on the eastern seaboard to replenish the money supply. The lack of money made selling or trading assets harder than it needed to be and was a check on US growth.

To finance the revolution, the former individual colonies, as well as the continental Congress, separately borrowed a great deal of money in any way that they could, as well as paying soldiers and others in IOUs. This lead to a great deal of financial uncertainty because it wasn't clear which debts would be paid on time, which might be paid slowly, or which might be repudiated and never paid. If one State defaulted, it could spur a crisis of confidence that any American debt would be repaid, crippling the government's ability to finance a future war or do any business for which gold or silver currency could not be paid immediately.

Hamilton's central bank scheme, the First Bank of the United States, pulled all the scattered debt into one single system and provided for an excise tax on whiskey to pay interest on the debt. Additionally, because the Federal Government owned the rights to a great deal of western land, it had a form of collateral to secure the debt.

However the true genius of the plan is that because the Federal debts were valuable to hold, and they were valuable because they paid interest and could be paid to the government in exchange for western land, they could be traded for gold or silver or goods similarly to how paper currency can be today. In essence, Federal debts were money. Thus they eased the cash problem inherent in the US larger US economy.

Of course, opposition existed based on the Excise tax itself, as well as the fact that a Federal government that could finance itself this was inherently more powerful. Finally, government control of paper currency and the banking system tended to benefit elites in Philadelphia and New York much more than southern or western farmers.

It's also worth noting that this scheme was not new in Colonial Finance. Ben Franklin had run a similar program described here in a previous post:

Source: Ron Chernow's biography of Hamilton.