Body and brain are modes of experience, like everything else. To feel and experience the world of everyday experience, we possess body, senses, mind, psychic instrument, etc. They are all “part of us”, they don’t exist independently of us.
The illusion that Deepak refers to is the illusion of separate existence. Sometimes that term illusion causes confusion in our scientifically-minded world, as we may ask “then what is illusion”? The answer is, it is also an experience! No?
I prefer Saivism which, like non-dualist Vedanta, accepts Awareness as fundamental but does NOT reject the world. It takes everything in the world as part of Siva, manifesting as Sakti (the energy of creation, maintenance and reabsorption, concealment and Grace). Saivism states that Maya (illusion) is the greatest and most un-understood power of the Lord, Siva. Rather than rejecting Her (and we should be aware of that impossibility and also because she hilds the key to our own emancipation) we accept Her as the Divine Creativity.