(Forgive me for expounding on a very common theme as of late. I'm in the midst of a new understanding of my own failures, and writing about them seems fitting).
Our hearts become hardened by sin and we begin to neglect that each person around us is our salvation, that Christ is everywhere, filling all things, and that he gives himself to us in Holy Communion so that we might recognize Him everywhere else.
Ultimately, the immense curse and spiritual danger of our sin is that it blinds us. In our gluttonous self-absorption we cannot see others as Christ. For if we were willing to admit that God was so near, so physically present as our neighbor, then we wouldn't have the courage to keep on sinning. If we were to realize that as vessels of Christ, our neighbors are holy, then we could not ignore them.
It is our selfishness that isolates us from others; our hearts harden; Jesus is ignored and once again placed into the tomb as the resurrection is forgotten. There does not anywhere in this world exist a merely "personal" or "individual" sin - all sin is a denial of God, thus it is a denial of every human being.
Whatever you did to the least of these, you did to me...