Why does Christianity claim to be the only true religion? From where does the idea that Christianity the only true religion come? Other religions claim the same. What is the basis for the Christian claim?
At one level, this question is easily answerable. 

Christianity is based upon the idea that Jesus is who He claimed to be.  And He said, in John 14:6 


  I am the way and the truth and the life. No one comes to the Father
  except through me.


And if Jesus is who He claimed to be, then Christianity must be the only true religion, because He said specifically "No one comes to the Father except through me".

At another level, there are other answers, but too large to fit in a single answer here.

If you're looking for external evidence, there's an entire field of study, known as Apologetics that seeks to provide a defense for the veracity of Christianity, its teachings, the identity of God, and the Truth of the Bible.  

There are volumes of information available (as well as counter-points arguing against the Truth of Christianity).  Put simply, there's much too much information to make your question answerable in this forum. A Google search for "Apologetics," however, is a good place to start.