Please explain me the principle of the vernier calipers.

The Vernier Caliper works on the principle that “the graduations on vernier scale are such that the length of n divisions on vernier scale is equal to n-1 divisions of the main scale”.

Generally, the vernier scale has 10 divisions. Thus, the total length of these divisions is equal to n-1 = 10-1 = 9 divisions on the main scale. This corresponds to 9 mm. Thus, each division of vernier scale is smaller by 1/10 mm.