What is Christian's burden in The Pilgrim's Progress? What was the reason for his burden to fall off when he saw the cross?

Christian's burden is related directly to the Biblical idea of human sin—it is the knowledge of his sin that burdens him. In the Bible (according to Paul, Christ, and many sources in the Old Testament), the knowledge of good and evil—that is, the understanding that our actions are evil—is what causes our destruction, and therein lies the difference between humans and animals. Because we know evil as evil and still choose still to do it, we are broken and sinful creatures. Therefore, Christian's burden is his knowledge of his own sin.
Upon reaching the cross, the straps which bind his burden to him are broken and the burden rolls away into Christ's tomb. This is representative of the Christian idea that by coming to faith in Christ—accepting his death in place of your own for your sin—the Christian becomes pure. In Christian theology, the Christian essentially trades lives with Christ, taking on his righteousness and perfection as their own and becoming unburdened by sin because they know they are forgiven and no longer guilty of it.