In some ways you've answered your question already. While it is almost inconceivable for nations in the modern world to go to war without convincing its population of the need for war, that is also precisely it: the modern world. The world of 18th century European conflict that Kissinger is discussing here is a very different place than the world we live in today.

It's very important to note that the modern 'nation-state' is a relatively recent development. The average citizen of 18th century Prussia or Austria would not view their relationship with their state the same was as a 21st century American views the United States. In many ways, political loyalty was very much a local affair. The average inhabitant of the lands we group together as "Austria" or "France" may have had a sense of a "Frenchman" being distinct from a "German", but would not have necessarily self-identified with the idea of a "France" or "Germany". They were more likely to identify with their local community and/or immediate rulers rather than anything else. 'International relations' as it were was less a matter of relations between "nations" but rather of relations between individual princes.

To go on a bit more precisely, there's also an assumption that in order to prosecute a war, you would have had to convince the civillian population at large of the righteousness of their cause. However, in the era you're asking about, of pre-Napoleonic and pre-industrial warfare, there's definitely a much lesser need to mobilise the home front as it were when compared to the famous large scale conflicts of the 20th century. The era Kissinger is talking about, where war is conducted on almost purely realist terms, is the time of Kabinettskriege, held distinct from the wars of religion that preceded them in the 17th and 16th centuries, as well as from the revolutionary wars of the 19th century. The armies in this era were almost uniformly--relatively--small and professionalised. While they were expensive to maintain, they also did not require a mass mobilisation of every resource of the state (not that states of the time could have done so even if they wanted to). After all, while a 20th century army needed a huge amount of equipment produced in industrial quantities, armies of the 18th century had much more limited requirements. There simply was not a need to create the kind of vast industrial production that characterises the World Wars, and so there was similarly no need for a mass mobilisation of the population.

With all this in mind, I'll try to sum up an answer for you: in many ways, the issue of how to convince the populations to go along with their princes is almost irrelevant. The populations in question would not have had much of a sense of "national" identity, so there wasn't much of a population to convince anyway. Relations at the times were less a matter of peoples, but a matter of princes, and the populations were relatively used to providing resources to their rulers. Secondly, it also wasn't necessary to do so anyway. You didn't need the kind of mass mobilisations we saw in the World Wars, or even in the Napoleonic Wars, without those requirements, you also don't need the kind of mass popular approval of war you see in those conflicts.

Hopefully this helps to answer your question. Feel free to ask any follow ups!