The Principal is just one person, so it is singular. The word "staff" usually refers to a group of people to begin with, so it is plural on it's own. The Principal (singular) and the staff are all waiting together, as a group of multiple people. This is why the sentence is plural.

Well, "staff" is already a plural word.

The principal and staff are grammatically considered a group of more than one person. You can prove this by reversing the two whereby you will notice that the verb stays in plural. Same as for example: Bob and Frank are happy. Bob and the cyclists are happy. The cyclists and Bob are happy.

As an AE speaker I've said both in this sort of case. However, when I'm careful I always say 'are.' Is' sounds déclassé to me. However, this observation by Su.Ki is correct: "Users of American English nearly always treat nouns such as staff, team, committee as grammatically singular."

We have read in grammar that if there is one article(the) in the subject, it acts as singular. But here it seems different. Please clear the qualm of mine.