Actually, all the answers so far assume too much. We have to recall that, in Washington's time, there weren't very many banks of any kind in the United States, and issuing government paper money was illegal.
The truth is that Hamilton, from an early time, openly favored the "monied" class and sought to woo them to support of the national government. He believed that having "people of quality" behind federal administration was the most certain way to insure stability and national unity. To this end, a bank of issue was a serious power to be reckoned with.
Banks of issue (with the exception of Bank of England, which eventually gained legal-tender status for its notes) could issue paper money, but it was not backed in any way by the government. (In the beginning, Federal Reserve Notes also were not legal tender.) The national bank was supported, however, by the government's deposits, and since those deposits were all but guarantied (the government always can steal your car if it actually needs it to survive), having a national bank was the next best thing to being able to print ones own cash.
Hence, being able to control a bank of issue gives the controlling authority tremendous power.
Banks of issue also are powerful instruments of war. Indeed, Bank of England was created for just that purpose in the 1690s. A bank of issue allows the Crown to fund its operations to some extent without having to rely on taxation (at least in the short run). This is a tremendous power and one Hamilton and Washington definitely would have appreciated, given their service during the Revolution. Since, at the time, no one recognized the flaws in embezzlement law (the first general embezzlement law was passed in England in 1799, and it applied to bank employees but not to banks, themselves), there appeared to be little downside to banking activities.
One problem was that the question of allowing federal corporations had arisen during the Constitutional Convention and been decisively voted down. The reason given was that those defending state interests did not want a national bank. They recognized that creating a corporation was a "great" power encouraging to monopolies and organized privilege.
So, Hamilton (and later John Marshall) had to come up with some reasoning which allowed him to get around the Constitution. He did this by asserting existence of a residuum of powers, unexpressed, which all governments inherently possess. That argument is weak, given the historical record, but one problem at the time was that the historical record still was hidden (Madison's notes did not become available until the 1830s). Later, Marshall acknowledged that "great" powers could not come into the Constitution incidentally but argued that corporations never could be an end of government and therefore, as means, could be included pursuant to the Incidental Powers Clause (another example of stilted Federalist reasoning which eventually led to the Jeffersonian Revolution and, in the case of Marshall, Jackson's retaliation).
At the heart of all of this is the question of who controls the bank. A national bank is chartered by the federal government and answers to organized "special" interests at the national level -- the "respectable" and "monied" classes which Federalists courted. State banks are controlled by local interests, and while they often were even more dangerous, they were seen as being closer to the people and therefore more democratic.
True Jeffersonians did not like banks at all, because banks at any level were viewed by them as engines of privilege for "connected" persons -- the very classes the Jeffersonians opposed. Hamilton, however, wanted something to spur manufactures and national commerce. He saw in banking operations ("financial schemes" to the Jeffersonians) a tool for obtaining that.
A National Bank would be superior to having individual state banks. A National bank gave more control over the issuance of money and credit. Trading policies with foreign nations would be done at one site instead of many sites. Hamilton did not want the same problem with the banks that the colonies had with taxes. A central bank is the only thing that prevented an economic disaster.
a, to absorb the debt, this was done by creating a bonds drive within the colonies and helped stabilize a very shaky economy. Hamilton believed in a strong and centralized government, excise taxes on alcohol and tobacco, and a tariff system on imports. the Revolutionary created a huge debt that the newly independant states would have to pay off. Buying bonds would help in paring down the debt.
A nation should have one currency used throughout. Only a central bark should be allowed to print money, thus controlling inflation. Trade could be more easily regulated.
