Germans usually learn at a very young age that a transaction among polite people requires two words: "danke" ("thanks") from the person who receives something, and "bitte" (roughly translatable to "you are welcome") from the person who gives something. So "bitte" is a really important word! Later they also learn that "bitte" can be used to ask for a favour. And yet later they may find out that many other languages have two different expressions for bitte and that one should use one expression during transactions and another one when asking for a favour, and never confuse the two (I am extrapolating a bit here from having gone through these steps myself).
Given that the verb "bitten" has the sense of "ask for a favor", I would assume that the original meaning of "bitte" is the same as "please". But how did "bitte" become such an important part of transactions? Is it an abbreviation for "please do not thank me", or for "please take this", or for both?