To be self-aware is to be able to conceive of something other than the present; i.e. to concieve at all, to view something in terms of something else - to not be wholely at the mercy of the ebb and flow of the present. By virtue of experiencing the flow of the present in terms of, through the lens of, some non-present experience, one achieves distance - i.e. self. To have selfhood is to be a subject which acts or is acted upon, to be selfless is to be action itself.
With some distance, one does something, something happens to one; i.e. there is an action AND an actor. Whereas, without any distance, there is a monism. One doesn't do something, something just happens, and then something else, etc. There is no distinction between actor and action - there is just action.
Nietzsche's concept of hierarchical physiology fits this view very well. He argued that their was in the body, as in all things, an order of rank. One could consider a relatively egalitarian - i.e. less complex - nervous system as more selfless, more 'in the moment,' lacking in higher-order 'deliberating' functionairies. Whereas, in a more complex nervous system, there is an oligarchy of higher-order functionairies managing the more automatic lower-ones: i.e. higher-order functionairies which 'deliberate' and gain some distance from the present. That is, the more complex the system, the less immediately affected are its highest order functionairies from whatever external stimuli are affecting the system as a whole - and thus, they gain the distance from the present, which in consciousness manifests as concepts, organization, intention, and other higher-order experiences: as opposed to raw sensation.
By this view, then, the difference between the self-awareness of a human being and that of animals lower down the list in complexity would be one of degree - as opposed to be some fundemental difference, like possession of a soul, etc.