Well, I agree with JLNobody on a couple of points. Firstly it's not a silly question, I don't think there is such a thing as a silly question when someone genuinely wants to know the answer.
Secondly, I believe that it is our ability to think abstractly which permits us to comtemplate the end of self. I would link this to thehamster's point that (as I am pretty sure) the evolutionary theory states that change happens in response to surroundings and situations, random genetic abberations occur which are found to be more suited to the changeing world. These changes are thus more successful (both in terms of survival and reproduction). Thus as the world changed, we changed. Some animals got bigger, some got faster, we got smarter.
Now it is unfair to say that animals are unaware of death, the contemplation of the death of self, however, requires that one look towards the future and have the ability to recognize chains of consequences (in other words you need a much longer attention span than your average animal

).