Through meditation we can certainly explore the 'Self' and recognise the Ego . Also, we know that we exist, but there is no good science to explain what our consciousness actually is.
LA,
It should not really be about explaining anything. We can possibly explain in some detail what we can see, hear and feel....because that''s what our brain and logic are evolved for.
Experiences and other abstract matters cannot be explained...but only indicated through analogies and allegories. This is why anthropomorphic ideas and myths come about.
For the vedantic ideas as in the OP, there is significant evidence...
1. As I have tried to discuss in the threads on 'Unconscious', 'Faith' and recently on the 'Living with hope...' thread .....the findings regarding the capabilities of the unconscious mind offer sufficient evidence that we have hidden inside us much greater capabilities than the conscious mind is capable of. This is indicative of a much larger 'self' than is apparent through the conscious mind.
2. NDE's are sufficient evidence that the 'self' lives on after death and.... in some cases there is evidence even of reincarnation.
No more evidence is possible nor necessary. As I have argued in the thread on 'Evidence'....everyone comes with a programmed (almost automated) response to any argument. It is not possible to change this response unless there is sufficient pressure on that person through direct experience. Arguments and discussions are not sufficient IMO.