First, he used the prosperity, or potential prosperity, of the new nation to sell bonds backed by gold. Next Hamilton advocated a tax on imports to gain revenue for the federal government. Third, Hamilton advocated subsuming the debts of the states into that of the federal government. This was achieved by political compromise with Jefferson and Madison to move the new federal capital to the south. Last, Hamilton instituted a program to refund the federal dollars in circulation at only 1% using a central bank established in 1791.
Hamilton's policies allowed the new nation huge new sources of income through an increase in tarriff revenue. This was especially true after Jay's Treaty in 1794. This treaty removed British forts from American territory and opened the Ohio Valley to settlement. Also Pinkney's Treaty with Spain in 1795 opened the port of New Orleans to American exports thus allowing the trade of 3/8 of the new nation to flow freely south down "the father of waters" to all parts of the world. Devasting inflation that ravaged the new nations of Latin America in this period was avoided by the funding of the debt.
So we see how political compromise, sound diplomacy, and stable economics put the early Republic back on a path to economic prosperity after the dislocations and devastations of the Revolutionary War. It was no accident. It was about the decisions made, the compromises reached, and the unshakable belief in the unbounded potential of the country and using that potential as leverage to fund debts, and play off the great powers of Europe against one another in a bid for our attentions and favors.