I wondered too. It might be that it could be used in queues in the sense normally expressed in English by next to indicate that a service position has become free (next person please). Not sure though - that's just a guess. A few recent phrases have seemed simply nonsensical or non-usable in real life and just used to illustrate a grammatical point.
"Next, please" and perhaps "Next in line, please" should be accepted as correct answers, IF that is what mtu mwingine tafadhali means. On the other hand, if it means, "Please bring me someone else because I can't deal with this person", or something similar, then that needs to be explained. We need a native speaker to tell us what the intended meaning is here.
As I understand it, the original word for the plural would have been "waingine" (with an 'i') but the 'a' and 'i' long ago elided into 'e', which gives "wengine". This is a common vowel change pattern in Swahili. I don't have a reference but I know I have read it in several textbooks.