Let H act on left cosets of H by left translation. The left coset xH is fixed by H iff every y in H produces yxH = xH. Multiply by x inverse on the left, and the conjugate of y has to map H onto H, hence the conjugate of y is something in H. This has to happen for every y in H, so the conjugate of H becomes H. The elements that conjugate H back into H form the normalizer of H, or in this case, K. Thus H fixes the cosets of H in K.
The number of fixed cosets is the index of H in K. By the fixed point principle, this index is congruent to the number of cosets in G, or the index of H in G. the index of H in K = the index of H in G mod p. Since the indexes are "congruent" mod p, this is called the congruent index principle.
If p divides the index of H in G, then p divides the index of H in K, and K properly contains H.