A kite flies in exactly the same way as an aircraft, thanks to its airflow profile (airfoil).
A kite flies when its lift is identical to its weight and it starts to pull as soon as the lift is higher.
As air flows over the aerodynamic profile, its speed changes. It is slower at the front edge and fastest at the widest section of the profile. The airflow decelerates until it reaches a speed of 0 at the rear edge.
With a symmetrical profile and no angle of attack, air flows over the upper and lower surfaces at an equal rate because there is no pressure differential. This means the aerodynamic profile doesn’t generate any lift.
If you now increase the angle between the airflow and the profile (the angle of attack), this causes a difference in pressure between the two sides. The overall lift is the lift for the upper side minus the lift for the lower side.