"I am" ,when we say this line we refer to our physical existence of biological form or body.
In true sense our existence is not just the aliveness of the physical body. In deeper state of meditation ,we can realise and feel that "I am" means not just the alive physical body but something within the body.
In our daily life , we have seen that when a person dies , the physical body reamins but something slips out of the body which was sensing its aliveness and saying "I am" when the person was alive. We are something more than body because our mind think about it.
Science says that our mind is the storehouse of the senses which we recieve from our sense organs in our day to day life. It records,analyze,calculate and give us the logics based on the past experience and it thinks in terms of the past only. It can not analyze in terms of futuer because it has no records of future.Therefore,our mind must be dependent upon the body and senses. Outer world and environment is influencing and shaping our mind.
As our mediation becomes deeper, thoughts slowly disappears and mind also vanishes. In that silence, we exist as soul.