The dative reflexive pronoun must be used when you specify which part of your body you are washing or which article of clothing you are putting on or taking off. Furthermore, the definite article is used to refer to the specific part of the body, wheras in English the possessive (my, your, his etc.) is used. It is therefore the task of the dative reflexive pronoun to indicate possession.
 
As the stem of the verb putzen ends in "-z-", note that the "du" form of the verb adds "-t", and not "-st" - du putzt.