What kind of information can we obtain from curvature for exemplar-based inpainting?
I haven't read the paper before, but the rationale is state in III.A:


  The filling order is very crucial which directly affects the quality of image inpainting. However, during filling procedure we find the data terms may be zeros for many pixels, which make the priorities being close to zeros, although the confidence terms may be very big. This would result in incorrect filling order.[...] by introducing the curvature term,
  we effectively avoid the priority being close to zero while the
  data term is zero.


The foundation for the method used in this paper is in Object removal by exemplar-based
inpainting, so be sure to read that first.