"Onegai shimasu" is one of those phrases that are a completely normal part of everyday Japanese culture that can sometimes be mystified and grandified beyond it's rather mundane usage.
First, a little linguistic background. As Jun mentioned in his linked post, "o + verb stem + suru (shimasu in this case)" is kenjougo, or "humble speech". One uses humble speech to indicate lower status, and/or modesty. However, "onegai shimasu" is so commonly used in everyday speech that it's actually moved to a different sphere. Rather than carrying connotations of modesty or lower status, it's more like teineigo, "polite speech". It's used in contexts when we'd use "Please", "Thanks," and "I'd appreciate it." For an example of the last two, my boss will frequently say, "You have to do A, B, and C. Onegai shimasu." My aikido sensei will say, "Everyone make sure you've paid the monthly dues! Onegai shimasu." Here the phrase has less literal meaning (sensei isn't saying "I humbly request your monthly dues,") it's more like a general softener tacked on to requests and orders.
Here's an example from our weekly meetings.
Manager: "Miitingu wo hajimemasu." (We'll begin the meeting.)
Us: "Onegai shimasu." (bowing in our seats)
Manager: "Onegai shimasu."
Here, again, the "onegai shimasu"es don't really have a specific meaning. We're not really saying "We humbly request that you do begin the meeting" (and the manager saying, "I humbly request your attention). The manager simply announces the beginning of the meeting, and our "onegai shimasu" is just an acknowledgment of that, and if anything marks a certain change in "kamae", from relaxing and joking before the meeting to being attentive and focused during the meeting. The manager's "onegai shimasu" is just a reflexive response (typically she's not even looking at us, but going over her notes or something). The meeting ends and everyone says, "Arigatou gozaimashita".
This is almost exactly the case in the dojo. Or at least my dojo. We sit in seiza, waiting for the sensei to begin the class. He comes over, kneels down, talks about some related business (upcoming testing, new member introductions, perhaps an acknowledgment of someone who's come for the first time in a while, etc.) and then bows and said "Jaa, hajimemasu. Onegai shimasu." We bow, say "Onegai shimasu" (except this is a dojo, so it's more like, "Ngai shmassss") and then right into tai no henko. Again, here they essentially mark the beginning of class, and acknowledgment of that and change of focus.
Likewise, the "onegai shimasu" between two students who are about to practice doesn't carry precise meaning (aside from maybe "Please be my partner"), and in my dojo, at least, is completely divorced from rank. Whoever notices the other person first is the first to say it, regardless of sempai-kohai. It's essentially an attention-getter/signal of change in attitude.
Another example outside of aikido: you'll see almost all interviews on TV/radio begin with mutual "onegai shimasu"es.