According to All Things Liberty, the American colonists financed their revolution through printing currency (both at the state and colony-wide Congressional level). Although some debt was issued as well, the primary funding came from printing currency. If the war was won, this currency would be needed to pay taxes to the new state and federal governments.
This is similar to the recent spate of cryptocurrency Initial Coin Offerings (ICOs). A group of people issue a currency to raise money for a venture. If the venture succeeds, the currency can be used on the new platform. If the venture fails, the currency will have no use.
There are a lot of problems with the ICO model, but financing a revolution with your own currency seems preferable to taking on external debt. Taking on external debt to finance a war could easily mean trading military occupation for exploitation by creditor(s). It is a good thing that the founding fathers printed money rather than borrowing from other 18th-century powers.
Happy 4th of July!
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