Why do Catholics worship the Pope?
"Worship" is neither the correct word nor the appropriate concept to describe the relationship between the core Catholic community and its institutional head. The Church is not "the Pope's Church." It is always the church whose beginning and whose end is Jesus Christ. This is an absolute. Everything Catholics do sacramentally and everything they should do institutionally flows from the real headship of Christ.
As a real community, a large one, making its way in the world, Roman Catholicism abides by a principle of authority that is the product of a very long tradition. The history of the papacy is an amazing story, and one that gives a face, positively and negatively, to the history of the Catholic community over the centuries. The "lineage" of the popes provides a focus around which the eras of that history can be given unique characters. Sociologically, this may help to explain why Catholics find special attachment in their hearts for the popes of their era.
Theologically, Catholics understand that the papal sequence traces back historically to the apostolic period. The New Testament places the authority delegated by Jesus to the apostolic community on Peter, who became the apostle with the greatest share of the accountability within the ancestral Church. From the very next generation Peter's successors in Rome were viewed as the primary "overseers" or bishops, of the growing assembly, "Church," which always saw itself as one with the bishop of Rome. And in that same city during its pagan imperial period, the more important efforts to destroy the young Church were focused on the Christian community at Rome. In a perverse way, the world itself acknowledged the importance of the "See of Peter" from a very early time. This diocese suffered like no other at the hands of the first persecutors.
Finally, we should note that no one mistakes the papal authority for the duty of the human conscience to bind itself to God's will. But by virtue of the greatness of the office in history, the sanctity of its origins and the weight of its mandate of teaching and leadership, Catholics today find in the papacy a special kind of authority. It is not the only contact or (necessarily) the most important contact that Catholics have with their Church, but it is still real and defining.