What is the basis of my fractured understanding of  "Sola Scriptura"? As a Catholic, I'm often baffled by other Christians who base their faith purely off personal interpretation of the Bible as the be-all-end-all proof for everything under the sun (which is what I mean by Sola Scriptura).  

I understand the need to figure out what's going on in the bible and the need to live your life according to wisdom of the prophets and saints and especially heed the words of Jesus. But where did the idea come from that everyone can interpret the bible for themselves?  Is this just a common Catholic prejudice or a misunderstanding?
As I understand it, the term Sola Scriptura isn't about personal interpretation so much as it refers to a closed canon, the notion that the Bible is complete and contain all necessary knowledge for salvation, and that no other words from any other source can legitimately be added to it and hold canonical status alongside the Bible.

As for its basis, there really isn't one.  Nowhere in the Bible does it say that the Bible is complete or closed.  (This is partially because the collection of individual works known today as "The Bible" wasn't actually compiled into its present form until centuries after the last of the individual works was written!)  The common justification that Revelation 22: 18-19 prohibits "adding to or taking away from" the Bible is clearly invalid in light of this, even before you consider that Revelation was actually written before John's other works, including the Gospel of John itself, and that extremely similar language is found in Deuteronomy chapter 4.  (Interpreted literally, that would invalidate the vast majority of the Bible, including the entire New Testament!)

Nowhere in any actual revelation or the writings of any prophet or Apostle does it say anything even resembling "Thus saith the Lord, my words are at an end."  The concept emerged during the Reformation as a counter to Catholic traditional teachings that were not found in the Bible and, in the Reformers' view at least, directly contradicted teachings that were found in the Bible.  It's a useful rhetorical device, but it is not itself a scriptural doctrine.

In fact, when you look at the history presented in the Bible itself, the concept of a closed canon is always associated with apostate groups.  The Samaritans and the Saducees both held positions that diminished or invalidated (to different degrees) the prophets after Moses.  Likewise, the Jews in the New Testament time refused to accept the new writings of Jesus's Apostles, and created the Masoretic Text (the basis for the modern Old Testament canon) as a closed work, stating that this was "The Scriptures" and nothing else could hold that status.  Sound familiar?