How do better power cables "work"? They:
1. have better conductors than lamp cord (I.E. bigger gauge, fewer stands, better metal)
2. have better shielding and/or geometry that cancels RFI/EMI from getting into the wire itself
3. or, more importantly, have better geometry/shielding that prevents RFI/EMI from leaking out of the cord and interfering with your low level audio signals in your nearby interconnects or the fancy circuits inside your gear
4. have better connectors that grip IEC and AC plugs firmly and better alloys that more efficiently transfer current.
5. are fatter, with fancier wrappers that make their owners feel special.
To respond to some of the comments above that after market power cords are either a bunch of whoo-ie, or that at least beyond certain simple structural improvements have no real impact, I suggest that there is a lot of room for design improvements that can especially affect #3 above, and that this is the place where the greatest benefits accrue.
I hear differences in sound between $99, $250 and $800 cords. I agree that the incremental benefits decay exponentially with increasing cost, but some people are willing to pay for those differences whether it is with wire, electronics, speakers and room treatments. If you don't hear any difference between different power cables, then God bless you and all that money you are saving.
PS - I could not help but laugh out loud at Stewie's posts above.