Oddly, it's often the most common expressions that are the hardest to translate literally. With such expressions, it's best not to try to piece apart each word, because often it doesn't make much sense. Instead, you can look at them as whole expressions.
Bitte can be used for a number of different meanings. It can be used to mean "please" (e.g. Können Sie mir bitte helfen? "Can you (formal) help me please?"), but it can also be used to mean "thank you" (e.g. if someone says Danke! you can reply with Bitte!).
Adding schön to create bitte schön still gives the meaning of "you're welcome," but it's a bit more polite. When you're using bitte to mean "you're welcome," you can think of it as a shorter, more casual form of bitte schön.
Like bitte, though, bitte schön can be used to mean things other than "you're welcome." If you are handing something to someone, you can say Bitte schön; in this situation, it essentially means the same as "Here you are / Here you go." You can also say it when you hold the door open for someone else to go through, in the sense of "Please, go ahead."
I hope this has helped to clear things up!