You don't transition from a barter to a currency economy, because there's no such thing as a barter economy. The textbook thought experiment about the value of currency as a solution to the mutual coincidence of wants is just that; a thought experiment, not an actual statement about history. Prior to currency was credit, not pure barter, and this is true of even the most ancient societies. (I find David Graeber generally obnoxious and he makes sometimes unbelievable errors, but about this, he has a point.) People may have done some barter, and indeed still do today, but there was never an entire economic system based around it. Since we have never observed a pure barter society, asking how such a society managed the transition to currency is an unanswerable question.
Ancient middle eastern writings and inscriptions contain almost super-abundant discussion of debts, repayment, forgiveness, amnesties, and so on, going back to ancient Sumer at the least. The Code of Hammurabi contains many, many references to debt, including much discussion about the circumstances of their repayment, and the unit of account, the mina, originally a unit of weight. The ever-popular Complaint Tablet to Ea-Nasir, the "world's first customer service complaint" from Babylon circa 1750 BC, contains reference to silver debts owed.
The threat of force underpins the whole system, as a centralised system of debts and obligations is only as good as the state(-like entity) that enforces it. But the basic idea is fundamental to any system of collection and storage, which necessarily involves accumulation at one time, and distribution at another. No barter need be involved, so long as there is an organization, governmental or religious, able and willing to keep track of these obligations. From there, the step to a common medium of exchange denominated in an ordinary unit of account (a token representing a weight of metal or grain, for instance) is a matter of degree, not a complete transformation in the economic system.
The gap between the emergence of these societies with centralized mechanisms for debt, and the development of coinage is over two thousand years. Metallic coins act as a secure medium of exchange and store of value despite the inherent fragility and untrustworthiness of states because they serve as their own insurance policy; however worthless the face value becomes, it's still a lump of metal that can be melted into something else. This solves the basic problem of how to get a currency introduced: make it an attractive store of value and a convenient medium of exchange. Precious metals are one fairly good solution to both problems, though certainly not without their flaws. Gold is soft and coins are easy to lose; gold and sliver do not have a consistent value against one another; coins can erode or be clipped, and so on. It takes another 1500 years after the introduction of coinage before the first society to issue what we could now recognize as sovereign fiat money, which is (AFAIK) China during the Tang and Song dynasties, though circulating private paper debts can be found much earlier.
The ways of getting a currency to circulate are many, but the basic problem is simple usefulness: if something serves the functions of currency better than the alternatives, then people will use it. If not, then not.
If a state wants its fiat money to circulate, then it must make it useful somehow,. especially as a store of value. That almost always means collecting taxes payable in that currency. (This is not *conceptually* necessary, as some crypto experiments from outside this subreddit's timeframe show, but it is by far the easiest and most popular method.) The taxes must be consistent enough to convince people that the money will be useful in the future, and that the supply of money will not increase enough to undermine its value. If people don't trust this, they will only accept it at a discount, proportional to how little they trust it. States that cannot continue to collect their taxes or redeem their currency in some other way will tend to see the value of their currency drop to zero, as in the case of (for instance) the Confederate dollar after the end of the US civil war.