Why do Baptists believe in the autonomy of local congregations? Why do Baptists reject the idea of a central authority and instead place authority in the local congregations? Does this have a biblical basis or is it a result of past conflict with authority figures?
There are two concepts that are important to understand.  How they interact show directly how this works:
Sola Scriptura
The idea of sola scriptura is that the Bible is the highest authority.  Anyone and everyone (who can read) has direct access to the final word of God.  If someone is saying something that isn't in line with the Bible, then they are clearly not from God.
Priesthood of All Believers
Essentially, every believer in Christ has the authority to preach, interpret the Bible, and determine doctrine (providing it falls in line with the Bible, of course).  Given this, the authority of Christianity lies directly with the believers in Christ, not with a church per se--neither a local congregation, nor a central authority.  Each person becomes responsible for his/her own beliefs.
Reaction!!
Now, combine those two ideas together and you get an emotional, fiery reaction:  All central authority crumbles, since each individual believer has the authority to reject or accept another person's word.  Furthermore, the only way to rebuke another believer to bring them in line with your way of thinking is using the Bible--a book that can be twisted, bent, confused, and misconstrued--all in the name of personal exegesis.
Local Congregations
As the dust settles, believers start to realize that there needs to be a church--a gathering place for others who share their beliefs--and a preacher--someone who can comfort, guide, and instruct them when they need answers.
Each church is built on these ideas: a place to gather for corporate worship, lead by someone to help guide, instruct, and lead the people.  When church leaders start to get off track, they are either let go, or the church falls apart.  The only cohesion being the Word of God and the beliefs established by the members of the congregation.
Authority of the Local Church
Finally, as all men do, the members of the congregation begin to stray.  Matthew 18:15-17 has given authority to believers to attempt to steer each other in the right direction.  If that doesn't work, some authority has been given to the church itself to guide members to right thinking: but which church?  A local church?  A central church?
Unfortunately, this passage (as with all) is open to interpretation; and schisms between believers throughout history have determined this to be (for their convenience, I suspect) authority of the local church, rather than a central authority.  That means the each local church has the ability to reject its own members.
However, if there are enough believers (which agree with the errant way) to form a new church, a new church is formed and a split occurs, dragging part of a congregation from one church (one authority) to another.
Summary
The doctrines of Sola Scriptura and the Priesthood of All Believers have caused each person to have the authority to reject or accept any given spiritual authority.  Churches band together on common beliefs for corporate worship, instruction, and support.  However, their cohesion is purely based on individual support of the church.
Ultimately, the authority is not with the local church as much as it is with the believers within that church.  Each church rises and falls on the beliefs established by its members.  It's these two doctrines that are the source of this &quot;authority&quot; of the local church (in a round-about way).