I think we can accept that weight is constant for a given moment in time. Lets also assume for the sake of simplicity that lift acts vertically upwards in opposition to weight. Thrust acts along the longitudinal axis of the aircraft. Now in a climb the aircraft pitch angle is angled slightly upwards which effectively adds some thrust to the lift. For the forces to balance we must reduce the lift to less than weight, so that the sum of lift and the thrust acting vertically now equal weight.