What is the importance of this quote in To Kill a Mockingbird?
The quote is "You never really understand a person until you consider things from his point of view-... until you climb into his skin and walk around it" (Chapter 3).
The quotation you mention is probably the most important in the novel, along with the line that includes "it's a sin to kill a mockingbird." The quote you mention is spoken by Atticus following Scout's disastrous first day at school. Scout has just told Atticus that she doesn't want to go back to school, what with all the problems she had: being punished by Miss Caroline, fighting with Walter Cunningham Jr., and then being scolded by Calpurnia. Atticus' suggestion to first climb into the other person's skin is a request for Scout to consider every individual's viewpoint before making a quick judgement.
Scout doesn't forget Atticus' advice. At the end of the novel, Scout stands on the Radley porch, pretending that she is standing in the shoes of Boo Radley. From this viewpoint, Scout sees her neighborhood from a different perspective--this time from the eyes of Boo. Things look differently to her this way, and she understands that Boo--Arthur--was just another neighbor observing the world around him.