Why is there extreme itching when we touch a hairy caterpillar? What are ways to stop itching?
It would seem that you're referring to caterpillars that have urticating hairs, such as the goldtail moth caterpillar.

Erebidae similis



  .. when we touch those hairs we get extreme itching on that area i want to know why we got that itching?


In this case, the itching sensation is the effect of a hair-specific, soluble protein known as Thaumetopoein. According to this publication, in where they studied the urticating hairs of the Thaumetopoea pityocampa moth caterpillar when applied to guinea-pig skin: 


  The hairs are hollow inside with sloping spikes toward the distal end. There are no holes or pores on the hair and the irritating factor can only be obtained by crushing the hair. This suggests that when stinging occurs the hair is broken in the skin and the substance [Thaumetopoein] is released.


More information about this immunological reaction begins on page 5 (numbered 350).




  And if one has got itching how to remove it? 



Remove all caterpillar hairs by first using tweezers to extract any embedded hairs, then the sticky side of tape to remove surface resting hairs. Do not use your bare hands, as this will only spread the caterpillar hairs.
Rinse the affected area with warm water &amp; soap. 
Apply some kind of hydrocortisone cream.


Poison.org - Caterpillar Stings