Suppose that you seat the $8$ people around the table in some order. Now each of them moves one place clockwise. No one is in the same seat as before, but they’re still in the same order: the arrangement has simply been rotated one place clockwise. If they repeat the manoeuvre, each of them will be in yet another seat, but they will still be in the same order. In fact, they won’t return to their original seats until they’ve repeated the manoeuvre $8$ times. There are therefore $8$ different seatings (including the original one) that preserve the original order and that are obtained from the original order by having the people move some number of seats clockwise. These $8$ seatings are equivalent under rotation: any one of them can be obtained from any of the others by having each person move a certain number of seats clockwise. One such set of $8$ seatings is an equivalence class of this notion of equivalence under rotation.
It may make things a little clearer to note that if two seatings are equivalent under rotation, each of the $8$ people has the same left and right neighbors in both seatings: everyone has simply moved around the table as a group.
There are $8!$ ways to arrange $8$ people in a line. If the seats at the table are numbered from $1$ through $8$, we can think of the seat numbers as positions in a line, and there are therefore $8!$ ways to seat the $8$ people around the table. But if we’re interested only in the order of the people around the table, rather than in their exact seat numbers, there are (as we just saw) $8$ different seatings that give the same circular order of the people. Thus, when we count these equivalence classes of rotationally equivalent seatings, we have to divide the total number of possible seatings by $8$. And the result is ... ?