Why do we wish "peace" to each other during Mass?

How did the practice of wishing the sign of peace to each other during Mass start? I only saw this after I moved here to the United States. I don't recall when going to Mass on Sundays back in Brazil, that the priest told us to wish peace to each other right after finishing praying the Our Father. Jesus didn't tell the apostles to do it! He gives us this peace, through the priest. Why do we need to give it to each other at the Mass? It doesn't make sense to me.