Let me have a go...
Let me define lift and drag as follows:
Lift is the force acting on the aircraft at right angles to the air speed vector.
Drag is the force acting on the aircraft at right angles to the lift, hence along the air speed vector.
In steady and horizontal flight, that means the lift points straight up, and is equal but opposite to the weight, which points straight down.
The drag points horizontally backwards along the line of flight, and is equal but opposite to the thrust, which is pointing forwards *)
In other words, the classic diagram we all know.
Now, what happens in a steady climb at the same speed?
The weight still points straight down.
But lift, drag and thrust now all have rotated over the climb angle.
Since we are talking about a steady climb, the weight is now balanced, not by the lift L (as defined before), but by Lcos(climb angle).
And that means that L has increased, which in practice is achieved by an increase in angle of attack (AoA, alpha).
And an increase in L means an increase in induced drag.
CJ
*) I've ignored all the second-order effects, such as the thrust line not being exactly along the line of flight, etc. which are not part of the basic question.