Thrust is needed to overcome drag, and a good airplane design can create a lot of lift for little drag. In the case of an A-320, the lift-to-drag ratio is 18 in cruise (a little less during take-off), so to lift those 78 tons needs only 4.33 tons of thrust. All it needs is the right amount of forward speed, and the engines only need to maintain this speed.
Gliders are even more efficient, their L/D can approach 60. A glider of maybe 450 kg mass (= 4412 N weight) will create a drag force of only 74 N, admittedly at a much slower speed than an A-320.
To be more precise, in a climb the thrust will actually compensate a little of the weight, in addition to the drag, depending on the climb angle. The thrust needs to grow by weight $\cdot$ sin($\gamma$), where $\gamma$ is the flight path angle. If the A-320 had enough thrust to lift all its weight, it could take off vertically, flying in a nose-up, vertical attitude. Modern fighter aircraft do really have that much thrust installed.
This drawing shows a climbing aircraft. There are four forces at work:
- Lift L (blue arrow)
- Drag D (red arrow)
- Thrust T (green arrow), and
- Weight m$\cdot$g (black arrow)
For a trimmed state, the concatenated arrows should form a closed trapezoid (lightly shaded arrows). Note that both thrust and drag are drawn longer here in relation to weight and lift than they are in reality.
To answer the why part needs a few more lines. Lift is created by deflecting the airstream over the wings downwards. This change in momentum is lift, and creating it would be free of drag if the wing's span would be infinite and the air inviscid. In reality, flying causes these drag components:
- Lift creation incurs a small backwards-pointing component which is biggest at slow speed and decreases with speed. To be precise: This drag component is proportional to the aircraft's span loading (weight divided by wingspan) and the inverse square of airspeed. It is called induced drag because it could be successfully calculated first with the same equations that describe electric induction.
- Flying through air causes friction. This friction drag grows almost with the square of airspeed and is proportional to air density and surface area.
- In inviscid flow, the sum of pressures around a body would just cancel out. In reality, the pressure on the forward-facing sections is normally higher than that on the aft-facing sections. This drag component is called pressure drag and is also proportional to the square of airspeed at the same angle of attack.
At the speed of best L/D the sum of 2 and 3 is equal to 1, and 2 and 3 are of the same magnitude for a well-designed airplane. High subsonic and supersonic flight adds a fourth component, called wave drag, and it's onset is the reason why the A-320 prefers to fly not much faster than Mach 0.78.