Electricity flows in a circuit. You have a wire going out from the generator to the load and another wire coming back from the load to the generator. If you have three circuits, you have six wires, three out and three back.
In 3 phase systems, each phase is 120 degrees off from the other. Three sine waves 120 degrees apart add up to zero. If you took your three circuits and tied all of the return wires together, and the loads were perfectly balanced, there would be zero current running through the return wire. If there’s no current, then you really don’t need the wire and you can get rid of it completely.
Of course, in reality you’ll never have the currents perfectly balanced. So what you end up with is 3 big wires and one little return wire, which is still a pretty big savings over 6 big wires.
3 phase also has an advantage for large motors. In a single phase motor, the torque “pulses” along with the AC current. When the sine wave is zero, at that instant there is zero torque coming from the motor. In a 3 phase motor though, when one phase is at the zero point in the sine wave, the other 2 phases aren’t, so there’s always torque on the motor. This makes for a smoother running motor (less vibration), which gets important when you are talking about very big motors.
If you need more power on a single phase motor, you need to increase the size of the two wires and the insulation to handle the greater voltage and current. For the same size wire, a 3 phase motor will give you significantly more power at a cost of only one extra wire. It’s basically 3 times the power with 1.5 times as much wire.
There are two types of residential electrical service in the U.S. By far the most common is that once the three phase reaches the distribution level, it is split out into three separate single phases. Transformers on the distribution line then take this single phase down to the power that you get into your house, so called “split phase” because they split the transformer with a center tap. The center tap becomes your neutral, and the two lines from either end of the transformer winding become your two “hot” wires. Line to line voltage is 240 volts, and line to neutral (from either “hot” to the neutral center tap) is 120 volts.
Much less common, but still in use in some areas, you get two phases out of the three. This still gives you 120 volts between either “hot” wire and neutral, but only gives you 208 volts from line to line. Since 240/208 is typically only used by things like electric ovens and electric clothes dryers, the only real side effect of the lower 208 voltage is that it takes longer for your oven to heat up and longer for your clothes to dry.