The common usage of "to realize" is captured in the Oxford American Dictionary's first two definitions: "be fully aware of; conceive as real" and "understand clearly." To realize something, in this sense, is to have it enter your mind, to acquire it as knowledge, to think it. "I didn't realize the mime spoke Spanish." "She realized that everyone in England understands Doctor Who references." "He was distraught when he realized the man with the mullet was his barber."
This is the manner of realizing alluded to in the song, albeit on a much more profound, existential level. But while the third definition of realize is to "present as real," as in "the story was powerfully realized on stage," this seems to fall under the fourth definition in the OAD, which is the most interesting to me:
"to convert into actuality; achieve." This is the definition used when we talk about someone realizing their potential.
So we have two main uses for the verb "to realize": 1) to understand and 2) to become.
The song "Do You Realize??" could be less eloquently rephrased as "Do You Understand??" But to realize something with regards to the other definition, as in to become something, as in to shed one's former self, is a process that happens all the time even though we only commonly pay attention to major milestones, such as realizing adulthood when you graduate college or realizing parenthood when that initial burst of terror rips through you in the hospital. The first step you ever took, the first time you got into serious trouble, the first time you knew pure confidence, the first time you knew you were very good at something, the first time you considered yourself an authority are all realizations. They required multiple understandings, but they are, themselves, becomings, realizations. And sometimes they float by us like another forgettable moment of life.
These realizations are occurring all the time. The psychosocial aspect of the human being has so many different dimensions to it (from the simple cognitive/emotional/spiritual breakdown to theories of multiple intelligences) that each person progresses in different areas at different speeds at different times. I fried fish on my own for the first time last week; that was a realization. Just as characters in works of fiction change throughout a work due to the challenges they face or how characters in role playing games level up different attributes through facing the challenge of adventure, flesh-and-blood human beings develop as well. All moments are unique but also you are a unique you in each moment, as per Heraclitus's thought that "you could not step twice into the same river." We don't always recognize it if there isn't a ceremony to celebrate, because our minds are all over the place and life is happening, but the development of the individual is a lifelong process that begins at your first breath and doesn't end until your last. In-between, there are many realizations, along with what you are able to realize about them.
Just being aware that the second definition of the verb "to realize" means "to become" is a powerful fulcrum for personal change. When I smoked cigarettes, I knew I wanted to stop. Eventually, I had feelings of hatred toward myself for smoking, for not having control, even for still enjoying it. But I didn't fully realize myself as a nonsmoker until my actions aligned with my thoughts.
Because you can think something about yourself, you can want it, have expectations about it, realize ways that you don't want to be, but these ideas and emotions are different than actually being that thing. That requires cracking open your old self and stepping through the cocoon into the new incarnation of you.
Just the idea that this second definition exists has been very useful to me, even though ironically it sounds awkward every time I try to use it in a sentence. "Today I realized myself as a being capable of frying fish." Bleh. Not gonna make the cut in a poem any time soon. But the idea of the word itself manifests the notion that personal growth, personal change, personal evolution are always inherent possibilities to the human condition.
Language, being one of the major organizational structures of our psychology, allows us far more conscious control of this evolution. We aren't always able to direct the ways in which we change as people, but having words greatly helps us. Words are like dusted-off fossils we have dug up from our subconscious, and through the words we can realize things, and we can thereby realize our realizations.