The right hand stands for the deeds, but not just deeds performed with the hand. It includes all deeds that lead us to break this commandment: going where we should not go; talking to who we should not talk to; as well as touching what we should not touch.
If the eye or the hand are about to give rise to evil thoughts, evil actions, stop them. Do something better. Christ shows the contrast between the Pharisees and his disciples. This was a standard of conduct that we completely out of reach for the Pharisees. There is surgery involved in the solution, but it is viewed as successful.