"If a climb is started from cruise flight, the airspeed gradually decreases is the airplane enters a stabilized climb attitude. The thrust required to maintain straight-and-leve flight at a given airspeed is not sufficient to maintain the same airspeed in a climb. Increase drag in a climb stems from increased lift demands made upon the wing to increase altitude. Climbing requires an excess of lift over that necessary to maintain level flight Increased lift will generate more induced drag. That increase in induced drag is why more power is needed and why a sustained climb requires an excess of thrust.

For practical purposes gravity or weight is a constant. A vector diagram shows why more lift is necessary during a climb, as the vertical component of lift generated from the wings is no longer perpendicular to the wings and adds to drag. The total vertical force is increased by adding a vertical component of thrust from the powerplant and the power should be advanced to the recommended climb power"

The above is what the FAA Airplane Flying Handbook says. However, I don't understand the bold statement. Why is more lift needed in a climb? Does it mean that more lift is needed to initiate a climb? Isn't it true that less lift is needed during a climb? Also I can't figure out how a vector diagram shows more lift is needed. I understand that the vertical component of lift is no longer perpendicular to the wings but it doesn't add to drag, does it? It is the horizontal component of weight that adds to drag, right?