It comes from "bergen", meaning "in Sicherheit bringen" (to shelter), as in "ein Schiff bergen", "wir verbargen uns im Schrank". From this stem nouns like "Herberge" and also "Geborgenheit". However, it seems the word "Burg" simply comes from "Berg" (mountain).

Of course I don't know this by heart, I recommend looking this up in

  https://www.dwds.de

Careful, you can easily spend hours there...