It's because religion allows them to do that. Keep in mind that religion is like an out of control book club where everyone has some idea as to what it is and constantly has to come back to the source to see what is says or to take what it says for their own interpretation. It's enough to where there are more who agree with a fringe idea than the rest of the book club is comfortable with and it just takes at least one of those fanatics to do something that will draw more attention to that fringe idea.
I like your analogy. You are right about the interpretation. Everyone bases their actions on their religious beliefs and their interpretation of the doctrine of that religious idea. However, two people reading the same doctrine are going to come out with different ideas as to its meaning. Not to mention, one may read an entire passage along with all the references that coincide with that reference while the other will read only part of a passage and nothing else. Thus they would only get half the meaning and not the whole idea.
That is the whole premises behind most Islamic Extremist and their ideals of rebellion, infidels, and jihad. Their idea's stem from partial reading and not from full understanding. I don't blame the members of the terrorist cells more so then their religious leaders because it is the responsibility of their leaders to read and fully grasp the concepts of their faith and present it to the people in the correct form. Also there is the problem of over interpretation. The is very simple and straight forward... Like command six in the ten commandments Thou Shalt Not Murder. It does not say not to kill... it says not to commit murder. Simple and straight forward, but many Christians call this commandment out whenever there is a war or when someone kills in defense of themselves, hell our laws against killing come from this commandment... but it does not say kill, it says murder. Murder is completely different from killing. A simple concept interpreted into a more complex idea then it had to be. Just my opinion.