As an adjective, bitter means difficult or distasteful to accept, admit, or bear, and it stems from severe grief, anguish, or disappointment; e.g., the bitter truth; bitter sorrow.
As for the meaning of Coffee remains bitter till someone stirs it--a saying, by the way, that is unfamilar to me, my best guess is this: if someone is bitter, they will most likely stay in that state for a long time. Unless, that is, they choose to forget the taste of bitterness by focussing on something else--like, say, stirring the coffee. (I could be wrong )
There's a similar saying, When you look into an abyss, the abyss also looks into you. ~ Friedrich Nietzsche
Does that help?