Peter was the first pope. This was established by Jesus in Mathew 16:18 when he promised to build his Church upon Peter’s foundation and when he gave Peter the keys. The keys signify, as we can see in Isaiah 22:19-23, the office of vicar, or royal steward, who has the king’s authority while the king is away.
Peter did many things, but he eventually became the bishop of Rome. Because the Bible is clear that Peter was specifically given the keys, it has become tradition that those keys are passed to the succeeding bishop of Rome. It’s that simple.
If you look at Church history, primacy among the bishops was always given to the bishop of Rome; both in the East and the West. As with so many things, we haven’t changed what the first Christians did. If it worked for them; it works for us.