Like all social change it tended to be cumulative rather than the new immediately replacing the old. It began perhaps twelve to fifty thousand years or more ago, and continues today. 

How is it that this, at first sight minor element of society, is the major factor affecting the organization and survival of Homo Sapiens and its cultures?

An important part of that is it produced a food surplus which allowed some members of society to produce the food and other members to concentrate on other things.  Perhaps more importantly, it required changes in our ways of thinking about the world around us, and those changes affected how we arranged ourselves

It facilitated and promoted many revolutionary social changes: urbanism and urbanization, writing and accounting, division of labour, concentration of population, and the formation of social classes based upon allocation of the food surplus (aristocracy, scribes, civil servants, accountants, military, police, traders, legal professions, medical practitioners, engineers, planners, infrastructure builders, trainers, food producers and the disenfranchised).
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Agriculture:
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The mode of production called agriculture means the human domestication of plants and animals. 
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The domestication of plants requires some saving instead of consuming all of the harvest, fruit and seeds, for the following growing season (leading to economic and religious ideas of sacrifice and investment). While gatherers consume (or store) what they gather, tillers put some aside to use for seeds the following season. This requires a new way of thinking about the universe.
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The domestication of animals requires controlling animals so they could be harvested when needed, are not dangerous to humans, and that their reproduction and offspring might be controlled also (equally leading to concepts of sacrifice and investment). This requires a different way of thinking than does hunting.
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Raising plants is called horticulture or tilling, while raising animals is called herding. True agriculture means the combination of both, even though, historically, these two modes were often incompatible; groups specializing in one were often in conflict with other groups specializing in the other. (Cain and Abel story).
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Thinking and Social Organization
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The new ways of thinking needed for both the two kinds of agriculture (plants and animals) were influential in changing social organization.  The idea of putting aside (to increase future production) instead of immediately consuming a harvest gave way to notions of sacrifice, saving and investment.
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How we think about things affects our culture and how we organize in society. Sociologists tell us that the very process of learning a language from birth shapes how we see reality. When we think things are a certain way, no matter how objective or not, for all intents and purposes they are that way.
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The Revolution Continues
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Agriculture continues to replace gathering and hunting, which do not call for human intervention in ensuring the supply of the product. 
The mode of food production is very important in shaping the social organization of a community, and therefore of the methods needed in community empowerment.

See keywords: Agricultural Revolution.