What if my micro-controller is on a breadboard? Let's say, I bought an AT90S8535 microcontroller chip only, how does this connect to it? Or a better example, an ATtiny?
and does the board give voltage to the microcontroller?
The six lines you need are:
SCK, MOSI, MISO, Ground, +5V, Reset
The first three are the SPI pins. The next two are power and yes, the USBasp provides 5V. The Reset is held low during serial programming.
You can find the 10-pin connector layout easily enough and run wires from the cable to your breadboard. If you eventually make a custom board you should include a 6-pin or 10-pin header.