Your question started me wondering too. The explanations on the web aren't very convincing. The idea that wing tip vortices are responsible sounds a bit odd and the argument that the efficiency of longer wings proves it would require the effects of wing loading (reduced circulation around the wing) to be considered.
The rotation of the total force vector implies that the axial force increases more quickly than the normal force but for an inviscid fluid I am unclear as to the physical source of the axial force given that pressure can only act normal to the surface. So the origin of induced drag relies on the viscosity of the fluid (well that it is not zero at least).
Perhaps using momentum theory where the force on the wing is the reaction to the force applied to deflect and slow the air, the separation of the boundary layer at the trailing edge, and the path this opens for air at the edge to be less deflected as it moves upward.
Wouldn't buy the wing tip theory except as an extra inefficiency but if that's the right book answer...........