The key discovery of human history is that minds are ultimately mechanical, operate according to physical principles, and that there is no fundamental distinction between the bits of organic matter that process thoughts and bits of organic matter elsewhere. This is called reductionism (in the second sense):
Reductionism can mean either (a) an approach to understanding the nature of complex things by reducing them to the interactions of their parts, or to simpler or more fundamental things or (b) a philosophical position that a complex system is nothing but the sum of its parts, and that an account of it can be reduced to accounts of individual constituents. This can be said of objects, phenomena, explanations, theories, and meanings.
This discovery is interesting because it implies that 1) minds, previously thought to be mystical, can in principle be mass-produced in factories, 2) the human mind is just one possible type of mind and can theoretically be extended or permuted in millions of different ways.
Because of the substantial economic, creative, and moral value of intelligent minds relative to unthinking matter, it seems plausible that minds will be mass-produced when the capability exists to do so. The moment when that becomes possible is the most important moment in the history of the planet.
Since reductionism is true, minds can be described according to their non-mental constituent parts. We then see that the current situation, involving a lot of matter -- very little of it intelligent -- is an unstable equilibrium. When minds gain the ability to replicate and extend themselves rapidly, they will do so. It will be far easier to build and enhance minds than to destroy them, and numerous rewards for mindcrafting. Thus we can envision a saturation of local matter with intelligence.
Kurzweil mentions that we will "saturate the whole universe with our intelligence" -- that is the most interesting and important aspect of Singularitarian thinking. In the long term, we should think not of the creation of discrete entities that behave as agents similar to humans, but rather massive legions of spirit-like intelligence saturating all local matter.
This intelligence saturation effect is more important than any other technologies discussed in the transhumanist canon -- life extension, nanotechnology, physical enhancement, whatever. When these technologies truly bear fruit, it will be as a side effect of the intelligence explosion effect. Even if incremental progress is made prior to an intelligence explosion, in retrospect it will be seen as trivial relative to the progress made during the intelligence explosion itself.