Sure, we happend to evolve it.
From last I read scientists are starting to think awareness and understanding come from a large brain. At first they thought it was something special to primates, however with Dolphins recently (well like a year) passing the self awareness test, they have rethought it, and now are leaning towards it being a product of a brain large or complicated enough to understand.
Also something else to add is that our ancestors and cousins (like neandertal man) seemed to have some basic awareness as there have been graves found where crude ceremonies were performed for the deads passing, like an ancient funeral.