Suppose we want to find the number of ways to make $n$ people sit in a circle. All the arrangements in which everyone has the same nearest neighbours count as the same arrangement.
The total number of arranging $n$ people is $n!$
In any of these $n!$ arrangements, we can rotate the arrangement $n$ times and get the same arrangement.
How do we account for flips though? If everyone has a person sitting directly opposite to then, then we can flip the arrangement about that axis to get the same arrangement. We can also rotate the thing $n$ times following the flip to get the same arrangement. But flips can only be done if people are sitting directly opposite. There are no flips possible if, say, 5 people are sitting equally spaced apart in a circle.