What does "circle each other" mean?

Solution 1:

At it's most basic, it's the physical manoeuvre of keeping a distance, and moving tangentially, creating a circle. It's easier to show than tell: here is an example from MMA.

It doesn't need to involve physical fighting, though. It's a "checking each other out" behaviour. You see cats and dogs do it too.

Lastly, this is often used as a metaphor for those early stages of a relationship or meeting where the people are warily checking each other out to understand the situation.

Solution 2:

In its simplest form it means that two people stand apart looking at each other and then move in a circular path around their mid point still facing each other. The move allows each to see how the other moves, or perhaps to find a better position to see the other, and it sometimes sets the scene for action that may follow, such as a fight or a gunfight.

More generally it acts as an analogy for circumstances where people take time to form an opinion about each other by cautiously revealing aspects of themselves to each other. It is a prelude to deciding something: whether to trust each other; whether to make a deal; whether to argue; and similar situations.