What are the commonly accepted reasons 'human progress' only really took off in the last 4000 years or so? Or is that a misconception?

The remarkable acceleration of technological and cultural development goes back a good deal further than that, and there isn’t really yet a generally agreed explanation. If tool-making dates back nearly two million years and cave-painting tens of thousands, why did it take anatomically modern humans until 10,000 years ago to develop agriculture or another 5000 years to use writing or the wheel, and then only a few hundred to get from the steam engine to interplanetary exploration? And why has average human life expectancy doubled in the last century after barely advancing over millennia?

Probably the most widely accepted explanation is deepening accumulation of cultural and technological advances, with each enhancing conditions for those to follow. But rather than linear progression the trend incorporates expansion characteristic of application of “general-purpose technologies” in which a given advance may produce multiple outcomes: just as the modern steam engine emerged to pump water from mines but in more refined form went on to power the revolution in mechanised manufacture and rail and early road locomotion, so writing expanded from listing mundane transactions to recording humanity’s intellectual heritage, and printing and later IT made learning and research available to millions.

Another approach (foreshadowed by the Tunisian scholar Ibn Khaldun six centuries ago in connection with urban development) looks at the growth of our numbers, specialisation and interactions:

When civilization (population) increases, the (available) labor again increases. In turn, luxury again increases in correspondence with the increasing profit, and the customs and needs of luxury increase. Crafts are created to obtain (luxury products). The value realized from them increases, and, as a result, profits are again multiplied in the town. Production there is thriving even more than before. And so it goes with the second and third increase. All the additional labor serves luxury and wealth, in contrast to the original labor that served (the necessities of) life.

(Ibn Khaldun, The Muqaddimah, translated by Franz Rosenthal, 1969)

More generally, while a village of a few hundred farmers may develop sophisticated responses to immediate questions of water conservation, soil maintenance or grain storage, it will never eradicate smallpox or send a probe to Mars. But as human numbers have grown, so has the range of our enquiries and exchanges, from local clubs of scholarly gentlefolk to journals, conferences and international research collaborations spanning the world’s universities. Nor can rising material standards of living and lengthening lifespans be left out of the equation, as a rapidly widening margin over subsistence offers ever greater numbers more time and means to access and contribute to a growing volume of accumulated knowledge.

Another perhaps more speculative way of considering it is that each successive innovation itself raised further challenges: as the readily-available nearby land came under cultivation, the need to feed a growing population forced communities to seek to improve less appealing tracts or increase yields; organising the work and resolving disputes necessitated an administration, taxes and record-keeping; and the need to know when and where to plant and harvest what crop brought forth science. The growing complexity of society and its activities calls for ever more solutions, and today's affluence is no exception.