Actually, USB uses four wires: two for powering devices (+5v and ground) and two for data (Data+ and Data-, it is a serial bus after all). A normal extension cable just makes all four wires longer, increasing the resistance and chance for electromagnetic interference. A repeater actually reads the data signal and rebroadcasts it. So in a sense, your computer is now seeing the repeater as the device, rather than the actual device itself, and the device is seeing the repeater as the computer!
"Since the reapeater does not have its own power source to give it a kick, it has to somehow uses the pre-existing power from the cable itself. Therefore, a repeater must have some kind of intelligence at the bulky encasing end to refocus and "amplify" the signal since brute force is not an option, right?"
The repeater is powered off of the same 5v line that your device is powered from, so it does have a power source. Rather than the analog amplification you are thinking of, the digital signal is just read and then retransmitted (or repeated) from scratch down the other end.
As Crashman said, there is a limit to how many bus-powered repeaters (those without an external power supply) can be strung along, as each repeater (along with the added wire resistance) eats some of the finite (500mA I believe) current avaialble.
Hope that helps.