If buses arrive exactly every ten minutes, it’s true that your average wait time will be half that interval: 5 minutes. Qualitatively speaking, it’s easy to convince yourself that adding some variation to those arrivals will make the average wait time somewhat longer, as we’ll see here.
The waiting time paradox turns out to be a particular instance of a more general phenomenon, the inspection paradox, which is discussed at length in this enlightening post by Allen Downey: The Inspection Paradox Is Everywhere.
Briefly, the inspection paradox arises whenever the probability of observing a quantity is related to the quantity being observed. Allen gives one example of surveying university students about the average size of their classes. Though the school may truthfully advertise an average of 30 students per class, the average class size as experienced by students can be (and generally will be) much larger. The reason is that there are (of course) more students in the larger classes, and so you oversample large classes when computing the average experience of students.
In the case of a nominally 10-minute bus line, sometimes the span between arrivals will be longer than 10 minutes, and sometimes shorter, and if you arrive at a random time, you have more opportunities to encounter a longer interval than to encounter a shorter interval. And so it makes sense that the average span of time experienced by riders will be longer than the average span of time between buses, because the longer spans are over-sampled.
But the waiting time paradox makes a stronger claim than this: when the average span between arrivals is $N$ minutes, the average span experienced by riders is $2N$ minutes. Could this possibly be true?