Well, you go to the post office with a couple of pictures of yourself and a copy of your birth certificate. You fill out the form and give it to the clerk behind the counter with the pictures and the document and whatever the fee is. The clerk sends it off to the processing center and about 2 months later your passport arrives in the mail.
Do you think a senator would have done it this way? I believe it is possible, even likely, that there is a special procedure for senators.