【申し上げる】１．「言う」の謙譲語。２．（補助動詞）動詞の対象に敬意を表す。In this sentence, it should be understand by the explanation2. To show respect to someone.

haha good question!(^^♪ Sounds samurai!

お願い申上げまする sounds like something you might hear in period movies set, say, the 17the century and is not used at all now, not even in the most formal situations. That said, I can imagine some people might say it in mock, over-the-top politeness as a joke between friends. お願いします is the usual form and you also sometimes hear お願い申上げます, which is a more formal and polite version of お願いします. Announcements over the PA system in department stores or on trains in Japan start with お客様にお願い申し上げます or something along those lines.

前者は目上、後者は丁寧