Tat tvam Asi or 'That thou art' is a Mahavaakya (Great Pronouncement or Great Saying) in Vedanta Advaita literature. It is originally mentioned in the Chandogya Upanishad as a dialogue between Uddalaka and his son, Shvetaketu. In any case, the reason this author has brought this up is to emphasize the simplicity and depth of the meaning behind this simple pronouncement. Just mulling over this Mahavaakya can be in itself all that a seeker needs. However, it is imperative for the seeker to understand the meaning of each of these words, the placement of the words in the sentence and how to approach the process of assimilation of the profundity of this statement.
TAT is That or the Reality or Brahman or whatever label you may want to give the eternal Reality that is self evident to all. THOU is the "you" or the "me". This "me" is also called the Jiva (pronounced as 'Jeeva'). What you mistakenly take yourself to be is this Jiva or the personality or the individual. At its very essence, the Mahavaakya is self explanatory. Note that this statement starts with a That rather than Thou. Meaning thereby that what IS, or THAT is already present BEFORE a 'Thou' (or a 'you' or 'me') arises. The Mahavaakya can lose its meaning if translated as "Thou are That" or "you are That", as is often seen in Advaita literature. Translating it in this manner gives substance to the feeling that there is actually a 'you' or 'thou' or 'me'. The seeker continues to think that he/she is an entity with some substance and independent nature and begins to analyze the statement from that reference point of a 'me'. Unfortunately, this just reinforces the concept or the sense of a 'me' or an 'individual', which defeats the very purpose of this statement. Of course, one could argue that this placement of words in this Mahavaakya is due to Sanskrit grammar and has nothing to do with the essential meaning of the statement or how this author has proposed that it should be interpreted. But remember that language is just a mode of communication and one can go on arguing endlessly about irrelevant topics thereby failing to see what is being pointed to.
The meaning of the Mahavaakya is clear when pointed out that before a thought of a 'me' can arise, I already AM. If one can just stick with the identity of THAT, then there is no question of identifying with what arises or comes after THAT. If one starts off as THAT and sticks to it, and then understands that what arises as an 'I-thought' (or a 'me') is nothing but just a thought, the meaning becomes immediately clear. All thoughts, actions, feelings, emotions, or anything else can BE, only if THAT is there to start with. THAT is always and eternally the Present. What can one say if the identity with THAT is established once and for all? One can only say that " I am every thing and no thing in particular".
For some seekers, this direct pointing out works wonders and hence this simple three word sentence has been labeled a Mahavaakya. As THAT, one realizes that he/she is nothing but just the Pure Awareness. The he/she cannot exist apart from THAT/Pure Awareness. Just ponder over this powerful pointer.