Obviously because of the string and the tail. How the string is tied to the sheet of paper (at certain angles) plays an important role in the successful flight of the kite. First, the sheet of paper is made stiff using thin sticks so that it does not bend due to the forces of the wind. Next, 3-4 small strings of equal length are attached. All this is to make the kite fly at an angle with the direction of the wind.
According to the laws of physics, any force acting on can be separated into two components – one horizontal and the other vertical. Generally the vertical component lifts the object (against gravity) and the horizontal component pushes the object (along the direction of the force). Here the force acting on the kite is due to the wind.
In the case of pieces of paper, the wind force blows them away as they are not secured anywhere. But in the case of the kite, it is held by the string in on e direction (a restoring force) and the wind is exerting its force in the opposite direction.
The net force on the kite is resolved into the horizontal and vertical components mentioned above. These force components depend on the angle of the kite’s axis to the wind direction. When the lifting force is sufficient enough, the kite begins to fly. 
At times we pull the string or run into the wind. These are tricks to increase the lifting force and make the kite fly higher. Some kites tend to rotate along the axis of the string. To prevent this rotation, a long tail is attached.
The flight of the kite may look simple going by the above explanation. But there is a lot more to it: For example, if we closely watch the flying kite we can see that it is very dynamic. It adjusts its position continuously depending on the movement of the air surrounding it. All this it does, may be because it wants to fly!