How did public debt become such a key institution in modern states?

I'm primarily writing from an early modern standpoint, though many of the developments I discuss continue into modernity. A lot more keypresses could be spilled on the topic than I have here, so feel free to ask further questions.

There are a number of causes for the increasing reliance of governments upon debt. For the sake of clarification: "country" in the pre-modern world primarily means the head of state, whether emperor, king, or lesser title; debts were primarily a personal affair.

The first, which might seem fairly obvious, is that the related rises of large-scale intercontinental trade, the money economy, and the complexity of financial relationships made such debt a possibility. Only with the rise of banking and "credit" can such debts be taken. This is a much longer story than I'm going to write, unfortunately, but bear this in mind.

Secondly, and just as importantly, changes in military technology, and the sheer growth in army size, drove governments to require vast sums of ready cash to equip, pay, and supply competent armies. Prior to the Military Revolution (think gunpowder and large standing armies), war was fairly cheap for monarchs. When you went to war, your lords and their knights would gather their forces and their levies, and join you on the field (unless they didn't). Armies were small enough, in most cases, to be readily supplied by foraging/pillaging, and the campaign season was rarely more than a few months.

This changed dramatically with the introduction of gunpowder, as now a common man, provided someone could arm him, was now roughly competitive with the knight. Army sizes swelled, as recruiters hired the bottom rung men of society, gave them arms, and put them on the field. At the same time, this all cost money. I mentioned before that feudal levies could refuse to muster for the king; the new armies, driven by coin, had no such limitations.

Dovetailing with this development, monarchs/state governments gradually developed strong government institutions, from large legal apparatus, to information gathering, to palaces (like Versailles). These all demanded significant expenditures of capital, but also increased the power and scope of centralized government.

Monarchs used these developments to become more independent from the goodwill of their nobles, but this came at a large financial price. This was a difficulty for kings, as a majority of the monarch's income typically came in the form of goods and services, rather than money. Peasants typically had little or no coin to spare, naturally, and nobles often had legal exemptions from certain taxes or forms of taxation, limiting the monarch's reach.

As a result, monarchs came to increasingly rely on the merchant class, who had access to money. These relationships worked in a variety of ways: as loans, typically given for particular wars or projects; through trading of resources, in which the merchants would receive exclusive access to a share of the product of the king's mines or fields (the Fuggers received almost complete control over the Austrian silver mines, for instance); and, increasingly but altogether insufficiently, through taxation.

tl;dr: debt is an important institution in modern states because the financial demands placed on the state, both internally and externally, almost always outstrip revenue. Since the states need the money in order to retain authority and power, people with access to money are typically able to make healthy sums from an institution which is relatively unlikely to default. Over time, the result is the accumulation of public debt; while the best economic times might be able to reduce the public debt somewhat, as in the US during the Clinton years, the concerted effort needed to seriously pay down public debt requires many more sacrifices than politicians, or the public, are typically willing or able to make.