The evidence for this view starts with Eastern (Indian) philosophy (Vedic Science). The science describes in detail the construction of the universe beyond our known physical plane/dimensions/level. This science was formulated by the saints/sages/mystics of the eastern world.
I've heard it said that Western Science concerns itself with the external world (like going to the moon) while Eastern Science concerns itself with the internal world (Who am I?).
The sages/mystics found after years and lifetimes of introspection that all is part of the one great conciousness.
Next question should be, why should we accept anything these sages/mystics tell us. First of all the consistancy of their testimony. But secondly, they would say don't take our word for it, explore your own conciousness and determine the validity of our teachings yourself. Now, us novices want the answer in under an hour
but the exploration may take years or lifetimes.
Getting back to your original question, I would also add the evidence the westerners mis-label 'paranormal phenomena' including near-death experiences, reincarnational memories, ghosts, spirit correspondences, etc.. In Vedic Science the word 'paranormal' would not really make sense as these phenomenas are normal in the universe-construct they present (with many planes/levels/dimensions beyond this physical one).