The same 4 forces of lift, weight, thrust, and drag exist in a climb but some have been tilted.
Both weight and drag are acting rearwards, trying to prevent the aircraft from climbing. These two forces can be combined into one resultant force acting rearwards.
In order for the aircraft to remain in a steady climb, an equal and opposite resultant force must be created by the combined effects of lift and thrust.
Although the two resultant forces are equal and opposite, a closer inspection shows that not all four forces are equal.
The weight can be divided into two components – one component opposite lift and the other component acting parallel to the relative airflow (the same direction as drag).
Since the component of weight acting opposite lift is smaller than the total weight, the lift is also reduced to match this component.
The rearwards component of weight is acting in the same direction as drag, effectively increasing the total drag. To balance the forces and maintain a steady climb, thrust must increase to compensate for this rear weight component.
Since the rear weight component is not actually drag (it is just weight that acts in the direction of drag), the thrust is now greater than drag.
A common misunderstanding is that lift is what makes the aircraft climb. In fact, in a steady climb lift is always less than weight and it is the thrust that makes the aircraft climb.
We can see from these diagrams:
The greatest climb angle is achieved when the difference between thrust and drag is greatest (known as excess thrust). Since drag changes when airspeed is changed, every aircraft has one particular airspeed at which thrust exceeds drag by the greatest amount – this is the best angle of climb airspeed.
Full power is used in most training aircraft for climbing in order to maximise the excess thrust.