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Looking at human history on that time scale, two key turning points stand out. One is the Neolithic Revolution and the other is the Industrial Revolution. In both cases, specific technological breakthroughs (especially in agriculture but also in other fields) meant two things. First, they allowed fewer person-hours to be spent per unit of goods produced. Second, they allowed the overall size of the human population to increase beyond previous limits. Those two dynamics taken together, productivity growth and population growth, supported one another and produced cycles of exponential change.
Multiple Neolithic Revolutions took place independently in different regions. They involved hunter-gatherers settling down and domesticating plants and animals to become full-time agriculturalists. Once this change was underway, other developments became possible like metalworking, writing and statecraft. A Bronze Age followed in many (but not all) cases of Neolithic Revolution across Eurasia, Africa and the Americas.
The Industrial Revolution was initially centered in England but diffused globally to such an extent that we can think of it as a single global event unfolding over a few centuries. The key developments in agriculture here were not new domestications so much as other kinds of technical change like greater and more effective use of fertilizers, more advanced breeding and husbandry, etc. This was more or less simultaneous with the harnessing of coal and later petroleum for industrial production. Unprecedented scientific understandings of physical and biological laws have been an important part of all of this.
What caused the relative stagnation before and in between these two revolutions? At the risk of over-simplification we can think of this in terms of the ecological "carrying capacity" of human populations. Given a certain set of technologies, a society will reach a certain limit beyond which it cannot grow. If population begins to surpass that limit, problems like famine and disease will reverse the trend. The result is a kind of relative stability. These two revolutions broke through the previous ceilings of population growth and allowed unprecedented kinds of societies to emerge. Normal incremental improvements are not enough to do that.
If you'd like to understand all of this in more depth, I strongly recommend the book Maps of Time: An Introduction to Big History by David Christian.