The idea that only Buddhism holds the keys to self-realization is dangerous, frankly. The Catholic Church want you to believe the same thing, that only the Papal hierarchy can direct you to God.
In short, I take these people's word for it because I see no reason to doubt it, just like any non-enlightened person takes the word of Buddha or Buddhist monks that they were/are enlightened. There are numerous instances of Hindus like Ramana and Nisargadatta claiming that their state of mind is identical to what Buddhists call Enlightenment.
If anything, it's Buddhism that has these crazy religious narratives about the cycle or rebirth and so on. Nisargadatta, for example, rejects all this stuff. God, reincarnation, is all conceptual fluff; ignorance. Because what is the "reincarnation" concept to the silent mind? It a disturbance.