Question: 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?

Answer: 
If Christ is indeed the **Divine Human** (fully God and fully Man) there can be no other, and thus no other religion could lay claim to one who truly knows God and can bring us to him because he **is** God.

Given that Christianity is the cult of said Jesus Christ, it would follow that if Christians were true to him, they would have a genuine connection to God, and not merely man's reaching up toward him.
As pointed out already frequently, the claim obviously comes from the bible. However whether that makes it a valid claim or not is interesting to ponder:

As most other religions, Christianity is self referential: By definition, you are Christian because you believe in Jesus, and since you believe in Jesus you are Christian. 

The same line of reasoning applies for pretty much all other religions as well. "I believe in Zeus and therefore I'm a Hellenist". "I believe in Tenri-O-no-Mikoto and therefore I am a Tenrikyo"

Most religions do have an exclusion clause the explicitly declares other religions and gods invalid. In Christianity that's, for example, the first commandment (second commandment in the Talmud) "Thou shalt have no other gods before me" . Interestingly, the wording seems to imply that other gods do exist, you are simply required to stay away from them. 

Once you make this fundamental believe statement, the rest follows easily. For a "Christian" the claim is true "because it says so in the bible". For a non-Christian the bible is irrelevant and the claim cannot be substantiated.