Onegaishimasu: "please, to share this practice..." That to me also means: "please, to share this moment". So it is an excellent honorific to use when writing letters, answering the phone, etc. I usually accompany it with a slight standing bow. It is a true way of bringing aikido into the world off the mat, which brings it to yourself as well.
This is exactly how I would see it! But then again off the mat it holds different meaning based on context and by no means am I an expert I have only lived in Japan 2 years but if I wanted to make it simple I would say that it means "lets do" and applied to different situations it would take on a different meaning. Josh is more informed on this level.