How can I use separate power source for the Arduino and the motors
That motor driver has an optional on-board 5V regulator. It can be enabled / disabled with a jumper.
With that regulator enabled you just need to connect a ground and your control signals (ENA/IN1/IN2 or ENB/IN3/IN4).
If you disable that regulator (remove the jumper) then you need to provide power from the Arduino's 5V pin to the +5V screw terminal.
The ENx pin(s) (after removing any jumpers that may be on them) connect to any PWM capable pin (this is the speed control pin). IN1/IN2 or IN3/IN4 connect to any IO pin since these are just used to set the direction or control the braking modes.  ENA controls the speed of motor 1, and IN1/IN2 control the direction of motor 1. ENB and IN3/IN4 are likewise for motor 2.