The transistor is used here as an electronic on-off switch. The only current the Arduino has to provide is that flowing through the transistor's base (which is very, very small). The current that flows through the relay's coil (and the transistor, via collector and emitter) doesn't "run back" into Arduino via the ground connection. It simply returns to the power supply connected to the coil.
I think you can connect "Relay Power V+" to Arduino 5V ouput. Don't remeber the max current value, though.