Protestants base their separatist views on the idea of what they call sola scriptura. This means that they see the Scriptures as the only authority to guide their lives. This, of course, is evidence of Protestants' total lack of understanding of what a Sacred Text is all about.
Sacred Texts are inspired by God. But they are not only Sacred, they are texts as well.
So what is a text?
In essence, texts are machines that form communities. They are means of communication, grouping together individuals, to form a whole. Non-sacred texts form communities that are qualitatively more than the mere sum of the individuals of the group. In this respect, Sacred texts are not different. But they add Sacrality - and they add it to the community. The community of those who write and to those who read the texts, is Sacred and Inspired.
Texts are written by people. Sacred Texts are written by Sacred People. Catholics have no problems in understanding that the Divine can incarnate itself into the Ordinary, by which the Ordinary becomes Sacred.
The writers of the Bible are testimony to this Divine Inspiration. They were people. People who formed a community. A Sacred community.
Such Sacred communities are called Churches. From the earliest days of Christianity, the Church was seen as a community of people, bound by the Sacred Word, incarnating God as a community, united in Christ, not as individuals but as Sacred Communities. It's extremely important to stress the sociality of the Church. A Church is not a mere reading club (as Protestant Churches want to be), it is a Sacred Communio.
Hence, the Church is the Incarnation of God. And this insight suffices to understand why Catholics view the Church as a sacred institution, carrying authority.
The Church recognizes the essentially social nature of people. I would go so far as to say that the Church, representative of God on Earth, puts the community before the individual (the Sacred history of Christianity is full testimony to this), and what's more, the social bond between people is Sacred as well.
To me, the problem of the Incarnation of Christ, is crucial to understand the differences between Catholics and Protestants when it comes to the idea of sola scriptura. Whereas the latter deny Christ's presence in people, Catholics acknowledge that Christ is incarnated both in the Text, and through the Sacraments, in the Sacred community as well.
Protestants are purists who are afraid of the idea that the Divine may manifest itself in the Ordinary (people, communities). Catholics are not afraid of the union of the earthly (the social) and the Divine. Hence, Catholicism is the true universal religion, binding people as a Sacred community, stressing that the social bond of the Believers is sacred as well.
[still working on this little text]
This is my pop-theological explanation of the non-sensical nature of a doctrine like sola scriptura.