First you need a 12V DC power source for your head unit. You can use a cheap car battery charger to convert from an AC house outlet to the DC that you need. You can connect that to the 12V connection on the head unit's wire harness. Connect the negative part of the charger to the ground wire in the wire harness.
Next you need to identify which wires from the wire harness go to yout speakers, hopefully that info came with your head unit.
One thing to remember though is that most home stereo speakers are 8 ohm versus 4 ohm car stereo speakers. What this means is that there is going to be very little wattage produced to your home speakers.