What makes the loss of a flap track fairing strongly increase fuel consumption on Airbus planes?
The fairings double as anti-shock bodies (or Küchemann carrots) and help to reduce shocks on the lower side of the wing. Below you see a wind tunnel test from the Vickers VC10 development where flow visualization using oil was used at Mach 0.9:



Wing without (left) and with (right) anti-shock bodies at Mach 0.9 (picture source). The light line on the left side indicates a strong shock which increases drag substantially over the situation pictured on the right where the flap track fairings have been used to smooth out the sudden reduction in cross sectional area from the rear-loaded wing airfoil.

Please read this answer if you are unsure why aircraft experience more drag as they approach the speed of sound. By stretching the aircraft body over a larger length, the gradients in the cross sectional area distribution can be made more shallow such that the air has more opportunity to make way for the aircraft.