The phrase darse cuenta means to notice or realize.
Literally, darse is to give oneself, and cuenta is account.
This is the most common way to express noticing something or someone (with people, it usually has a romantic connotation).
Here are examples:
¿Ya te diste cuenta de la noticia? –> Did you notice the news?
No se ha dado cuenta de mí. –> He hasn’t noticed me.
No me di cuenta que tenía que amarrarme los cordones –> I hadn’t realized that I had to tie my laces.
You can see it in these videos: