So, it seems odd to most Americans -- and dare I say, most American Catholics -- that anyone would submit to the authority of a Pope in Rome or a collection of bishops. But if you follow the logic, you'll see that it's not so crazy, from the Catholic point of view. Think about it:
If Jesus literally rose from the dead, then He is God, and I will submit to Him in all things.
If Jesus established a Church, then I will be a member of His Church.
If Jesus appointed leaders for His Church and delegated His teaching authority to those leaders, then I will submit to those leaders when they speak on matters of salvation, faith and morals.
Those are some big ifs, and I won't set about to prove any of them today. Obviously, many folks dispute one or all of them.
If any one of those ifs is untrue, then the Catholic Church is irrelevant and can be ignored.
But if they all are true, then we have just answered the question, "Why does anyone care what the Catholic Church says?"