Most Christians interpret that scripture as Jesus explaining that only God can judge man, that other human beings are not qualified to judge each other. In a similar scripture Jesus warns his followers not to fear men who can harm the body but to fear God who can throw them both body and soul into Hell.
Also these teachings emphasize that all people have sin in their lives and since we're all lowly sinners we shouldn't go around casting stones. This can be illustrated in the story where Jesus stops the stoning of the adulteress as well as when he tells his disciples not to focus on the "speck" in your neighbor's eye when you have a "log" in your own. Rather than show moral relativism I always interpreted these teachings as being ones of self-responsibility, being responsible for your own sin and not the sins of others.
Unfortunately that idea of self-responsibility for sin is directly contradicted by Jesus taking on everyone's sins on the Cross. Christianity is filled with those sorts of contradictions, such as the battle between whether its faith, works, or some combination of the two, that lead to salvation. Plot holes like these have led to a lot of internal debate and help explain why there are so many divisions in Christianity.