This is "tat tvam asi"
(sanskrit) from the Upanishads, which represent one of the prevailing doctrines in the time of the Buddha. In most basic terms it relates "thou/you" with some eternal divinity/God/Brahman.
You can think of it as the reaction to our everyday notions about reality, where there is an "I" walking around in some world consisting of things (outside ourself). Mysticism in general (Hindu, Sufi, some Christian Mysticism) essentially says that there is no such separation and that "everything is one," or "we are one with God." The concept of enlightenment in such traditions is that you stop identifying with a small "self" and identify with the totality.
Buddhism is more like "thou art not that." You can think of it as a reaction to the above, or such a doctrine taken one step further. Instead of saying that there some kind of "self" that merges with an "other," Buddhism says that it's the identification with an idea of "self" that causes suffering. But this includes any concept of self whatsoever, including a cosmic or divine one.
You can roughly identify the concept of "thou art that" with Eastern philosophies, except for Buddhism, which teaches no-self or "anatta," instead. Hopefully your eastern philosophy teacher knows this, but a surprisingly large number of people do not, and lump all eastern philosophy together as teaching this sort of philosophy.
As for any personal interpretation of the meaning, I feel that anyone who can identify with some sort of cosmic, universal self, divinity, etc. then they are what I might consider an accomplished mystic or spiritually advanced or whatever, but they do not have the level of realization that the Buddha had, since he went much further to drop even this notion of a self.