If you'd asked people from millennia ago up until relatively recently about forms of transport they use, it's likely they'd have said something like horse and cart or sailing ship. But if you'd asked the same question from, say, 1800 to now, the answer could have been steam train, steam boat, car, plane, huge numbers of new inventions. Up until 50 years ago or so computers would have barely existed, now they're everywhere. People had been fighting with generally unchanging swords and bows for thousands of years, but just in the last hundred years machine guns, assault rifles, tanks, rockets, missiles, jet fighters and nuclear bombs have all not just been invented but become widespread.
It seems to me like since the Industrial Revolution the pace of technological advancement, sadly for reasons which appear to largely revolve around finding new ways to kill people, has become much faster. Is this the case? And if so, why? Does the rapidly increasing world population have anything to do with it?
It's important to remember that the idea that life was "unchanged" for thousands of years is not quite right. It looks that way to us because we often don't take the time to study it in detail or in depth. Things have always been changing. Yes, there are some common threads. But if you dig into the last 10,000 years of history (basically the rise of human civilizations) you find that all over the place there are vast changes all the time, including technological ones.
The reason they look relatively "static" compared to our current age, however, is that there have been real accelerations of technological developments in the past 200 years. There are many factors for why this has occurred:
the rise of scientific modes of thinking and a scientific profession that allows for the rapid development of underlying models of nature, out of which technological improvements sometimes come
the creation of systems of financing that allow for resources to be selectively concentrated in areas of interest
the rise of larger, more "modern" states that allow for the rapid accumulation of capital
the perception (since at least the mid-19th century) that technology plays a major role in the outcome of wars (the big "wake up call" for most governments was the Franco-Prussian War of 1870, but there were other indications earlier than this)
the development of large-scale industrial modes of production that both research and develop new technologies as part of their normal operation, and can be converted to military purposes as well
and since WWII in particular, the rise of military-industrial-complexes that apply unheard-of resources towards developing "cutting-edge" technological developments
And so on. You could generalize this by saying that starting in the mid-19th century, the trends were to increase the power of state governments, increase the scope of industry, and combine both of these with dedicated systems of production of knowledge to deliberately and quite consciously try to increase the number of breakthrough technologies. It is a matter both of scale (in terms of resources devoted) and mindset (the development of new technologies is a thing unto itself today, whereas it would have been a rather niche idea 400 years ago — consider the difference between the engineering school where I teach, which brands itself as "The Innovation University," and the musings of Leonardo Di Vinci in his secret notebooks).
But it should also be noted that this kind of generalization is always problematic — you should not see it as either a discontinuous "jump" (from one way to another) nor should you see it as a teleological development (the history of the world is not people learning to be "modern"). There are lots of trends and practices and developments that take a long time to unfold. We think of ourselves as being at the "peak" of this development but in terms of revolutionary technologies (or scientific discoveries) this is not necessarily even true (there are those who have argued that by the late 20th century a lot of the "low-hanging fruit" developments were figured out and much of what we have today are just tweaks and refinements as opposed to revolutions; there are both pros and cons to this way of thinking).
As for war being a factor — war is a factor inasmuch as states are a thread and states at times either fear or desire war. States direct resources and perceive technological advantage as aiding in war aims. So a large percentage of resources has been directed at technologies with specific military relevance. Again, this connection goes back quite a long way, but accelerates rapidly in the 19th and especially 20th centuries along with the increased role of centralized states and the increased perception of the value of military technology and the combination of industrialized production with military demands.
As for the connection to population, there have been some studies on the role of population clustering as a necessary prerequisite for some of these activities — large population bases provide haystacks to find talent in, they provide support to research both in the forms of tax revenues and commercial audiences, and there is probably a minimum population size to be able to support something like a thriving scientific professional, military-industrial complex, etc. But that being said, the minimum population of these things is probably much smaller than the current population; there are no doubt diminishing returns created by the increased needs of such populations, the sorts of economic conditions that create massively exponential population growths, and so on.
All that being said, one should also avoid thinking solely of the "new" technologies as defining our world — we live in a world that is a constant mingling of the new and the old. (I type this on a MacBook Pro connected to WiFi and communicating over the Internet, to be sure, but I do so while sitting on a wooden chair at a wooden table with a ceramic mug full of coffee, with a dog at my feet and a bookcase full of paper behind me.) The distinctions and differences between the "new" and the "old" can be overemphasized, and the impact of technology can be as well.
There are many further things one could cite. If I had to pick one that you might enjoy reading, I might suggest David Edgerton's The Shock of the Old: Technology and Global History since 1900, which is about the ways in which technological developments have indeed accelerated over time but their impact is often widely misunderstood. If you are interested in the connections between technology and war more generally, John Keegan's A History of Warfare is an enjoyable read that tries to synthesize many different sorts of ways in which technology and warfare have been co-influenced over the centuries (among other themes). For somewhat more heavy sledding, Bruno Latour's The Pasteurization of France is a discussion of what it takes for a laboratory to exist and be successful (it is more than having a smart person at the helm, or discovering true things — it is about the creation of a social system that allows authority and power to move from the laboratory into the broader world), and his We Have Never Been Modern is about the construction and critique of the "modern" mindset that powers and enables all of this technological activity.