They borrowed.
First, the individual states and the Continental Congress printed paper money, and as expenses exceeded income, they printed more of it. As a result the paper money depreciated in value to little or nothing, which eventually led to the phrase "not worth the Continental". This essentially was a tax levied on, money borrowed from, all the people who took the currency in exchange for actual goods- food and firewood for the troops, cloth for uniforms, etc.
Second, there were wealthy patriots who loaned the government money, like Oliver Pollock and Robert Morris. They often loaned the government money to buy their own goods. This would later lead to claims that they were war profiteers, hoping to eventually make huge amounts from these sales when they were repaid. But for the most part they were motivated by devotion to the cause.
Third, they borrowed from the French. That French navy outside Yorktown, blockading the British, was not free. and the French contributed both money and materiel to the colonists. When Morris started the First Bank of the United States, in order to begin to pay down the war debts, the French supplied the start cash.
At the end of the War the public debt was $42,000,375. $7,885,008 was foreign, $34,115,290 was domestic. Efforts were made to try to pay down the debt, but often at a discount. Continental currency was redeemed at 1/40th of the face value. Whether they thought they might profit from the Revolution, most all of the financiers in the end suffered for it. Pollock never recovered his business. Morris entangled his own finances with those of the Revolution to a dizzying extent, loaning the Army almost 14 million dollars, took a great losses. The repayment of the French with yearly tobacco shipments didn't last, and their unpaid loans to the Continental Congress and contributions to the American Revolution would be a significant part of the huge debt burden that finally collapsed their government and led to their own Revolution.
It should be noted, though, that part of Washington's success had to have been due to his ability to keep an army in the field despite lacking reliable funding. It was something he was forced to learn to do in the early part of the French and Indian War ( or Seven Years War) when the colonial governments were equally unwilling or unable to provide adequate financial support.
Anderson: Crucible of War