When you do something punctiliously, you pay attention to doing it carefully and precisely. During the changing of the guards at Buckingham Palace, British soldiers leave their posts punctiliously, following the rules of the ceremony every time.
Think about the extreme care taken by someone leading a Japanese tea ceremony: they bow, whisk the tea, and pour it, following each step of the ritual precisely and punctiliously. This adverb comes from punctilio, a "fine point," or minor detail. Paying attention to these small rules and details is what acting punctiliously is all about.