Onegaishimasu always reminds me of a phrase I learned when my family moved to a town in northern Finland when I was 9. Kids up there will ask each other, on the playground or coming to the front door "Will you begin me?" ("Alakkonää mua?") and yes, it's sounds just as odd and grammatically incorrect in Finnish.
Anyway, what they mean is "will you come out and play with me?" and that's the same feel I have with onegaishimasu when I bow to my partner.