Why would anyone believe a church? Any church needs to first establish its credibility, before it can hope to assert any authority.
The Roman Catholic Church, being the oldest surviving Christian church, claims direct lineage from the apostles (implying descent from Christ himself), more specifically the apostle Peter, and not only that, they (maybe not all) claim that pursuant to Matthew 16:18, only the lineage of Peter stand to inherent the entire church.
Beginning with Martin Luther, groups began to break away, and they got excommunicated by the RCC. They could no longer claim continuation of the lineage by the RCC, since they have been disowned (i.e. disinherited). And so they need to have their credibility established by another way, which they found with faith, the Bible, and other articles of faith (i.e. everyone goes directly to God).
There is a third claim to credibility, and this time, it is based on revelations by modern day prophets like Joseph Smith and the LDS (aka Mormons) or by modern day messengers like Felix Manalo and the INC. There is an implied lineage in these churches, beginning with the founder. Unless banned by claims of someone being the last, there is also an implied direct connection to God by anyone, if he or she is made of the stuff prophets and messengers are made of or simply by having been chosen by God or Christ to do so.
These are perhaps oversimplifications, though it may hopefully serve some useful idea.
I think behind the issue of Sola Scriptura or "by Scripture alone", would be the issue of credibility, which implies the issue of authority or legitimacy. It may yet be another form of the "wars of the churches."