If the unconscious exists as part of our working brains, how can we tell what is in it? Can we find out what is in specifically our own unconscious by ourselves?

There are different theories about that but one prominent theory, namely, Freud's - that repression and resistance are the reasons why much of our mental life is unconscious and save for himself -- he thought ordinary human beings could not break through that resistance on their own. As usual, Freud overstated things a mite, but there is something to be said for the view that we need some help in understanding the meaning of what is beneath the waves of our conscious lives - but that joint effort requires the ability to tolerate vulnerability and anxiety.