First off, Ray Kurtzweil IS a nutter. Trust me on this.
Secondly, despite being a nutter, he's not off by much. Assuming Moore's law holds true, we should be able to manufacture computers capable of emulating the human central nervous system in realtime in about 10-15 years. Every 1.5 years after that, assuming that existing design trends hold true, the speed of that computer will double, and the cost of the original computer will be cut in half.
Now, let's say we copy a person into one of these things, and set them around to designing computers. It now takes them 1.5 years subjective time to double the speed of a computer - which is 9 months realtime.
The next generation takes 4.5 months.
Then 2.25 months
Then 1.125 months
I think you see where this is going - eventually, this doubling of computational power runs away to infinity, or at least as fast as manufacturing processes can crank it out. Some people will say something along the lines of "what about the inherent physical limits on computer manufacturing technologies?" Well, that doubling isn't just about the physical limits themselves - it's also about figuring out how to manipulate the equipment we have. For example, current computer processors spend roughly 90% of their real estate making sure all calculations occur simultaneously with the clock phase drop. One idea that has been played with is the manufacturing of a "clockless" chip, which (if design problems were solved) would free up a huge amount of space on your average processor. There are also efficiencies to be gained in redesigning the (simulated) brain - a lot of "excess" baggage could be trimmed out. Evolution is great at designing something by piling lots of stuff on top of existing systems (we have 3-4 distinct parts of our central nervous system, with lots of redundancy built in, some of which does not get used for pretty much anything at all), but removing a lot of the cruft could be useful for greater operational efficiency.
The end result would be something not quite human - and we can't predict it's behavior. The only thing we can predict about the singularity is that things are going to CHANGE. Our own survival, either as individuals or as a species, is not guaranteed.
Read
Accelerando! for one example of what can go wrong with "Transcendance" - although, IMO, Economy 2.0 is already upon us...