You pose and interesting question, "Where did the idea come from that everyone can interpret the bible for themselves?"
An equally valid question is, "Where did the idea come from that any believer cannot interpret the Bible for himself? Every believer has the Spirit of God dwelling within him (1 Corinthians 6:19), so why would that believer have to be dependent on others, specifically a certain group of individuals, to understand the Scriptures?
The real question then is in determining where the burden of proof lies. Does it lie with those who claim only a select group of believers have the authority to interpret Scripture, or does it lie with those who claim any believer, within whom dwells the Spirit of God, can read and interpret Scripture.
Arguments for Sola Scriptura
The Bible written to All Believers
Most of the New Testament epistles are addressed not to clergy, but to all believers. The Gospels seem to be written for the benefit of both believers and unbelievers. From the earliest times of Christianity, the gospels and epistles were copied and passed on to be made available to a large audience. Therefore, we conclude that the Bible was intended to be read or heard by all people and not just a select group. Therefore, it would seem counter-intuitive to assume that all people were given the Word of God to read but then prevented from interpreting it.
"...To the churches of Galatia." Galatians 1:1
"...To the saints who are in Ephesus" Ephesians 1:1
"...To all the saints in Christ Jesus who are at Philippi..." Philippians 1:1
"To the saints and faithful brothers in Christ at Colossae..." Colossians 1:2
The Priesthood of All Believers
The New Testament also teaches that God has made all believers priests, and that there is no mediator between God and men except for Jesus Christ.
But you are a chosen race, a royal priesthood, a holy nation, a people
for his own possession, that you may proclaim the excellencies of him
who called you out of darkness into his marvelous light. 1 Peter 2:9
For there is one God, and there is one mediator between God and men,
the man Christ Jesus. 1 Timothy 2:5 ESV
So, if there is no mediator between God and men anymore, then there would be no need for an interpreter between God and men either.
No Biblical Teaching for Restricted Interpretation
Also, there is no place in Scripture (at least of which I am aware) that specifically restricts the interpretation of Scripture to a select group. If it were really the intent of God to restrict the interpretation of Scripture to a select group of people, then He would certainly have made that blatantly obvious in the Scriptures themselves. Yet, since there is no such teaching in Scripture, the source for that teaching must be extra-Biblical itself.
Problems with Limited Authority for Interpretation
Assuming that God did, in fact, restrict interpretation of Scripture to a select few, then it seems the purpose for this would be to make sure error and heresy never crept in to the church. The result would be that throughout church history, God would always work through the select group of people to maintain sound doctrine.
However, when we look at the Catholic church, which holds to this doctrine, we do not find a consistent teaching throughout history. Many doctrines have changed, including the celibacy of the priesthood, indulgences, the language of mass, eating meat on Fridays, etc. So, if the teachings have changed over time, then it would seem that the leaders wrongly interpreted Scripture at various times through history, unless we hold to the idea that truth can change. Therefore, the basis of limited authority for interpretation is undermined by the variance in teachings by the select group over the 2000 years of church history.
So, 1) since the Bible is given to all people to read and hear, 2) the New Testament teaches the priesthood of all believers, 3) there is no specific teaching in Scripture that would restrict interpretation to a select group of people, and 4) history has shown that the teaching of the select group has been inconsistent and contradictory, the conclusion seems reasonable that all believers have the authority with the indwelling Holy Spirit as a guide to read and interpret Scriptures apart from any mediator or interpreter.