This is an interesting Maxim in that we should not flee from our enemies but to flee from enmity itself. We should do what we can to avoid incurring enmity in the first place. This is not always easy as we travel through life and encounter other people, their actions, and their feelings. We bump into those things on a regular basis and it is our reaction to them that determines how well we are able to navigate this Maxim.
Do we allow anger to dictate our actions? Do we allow hurt feelings to dictate our actions? Do we act on partial information only to regret it later? These are questions that we must ask ourselves on a regular basis as we maneuver through day-to-day life. These are questions that need to be answered before we act on hurt feelings and anger, causing enmities which can poison relationships for long periods of time.
So how do we avoid falling victim to enmities? First, we must honestly answer the questions about how our own feelings can affect our reactions. Then, we must act in a way that ensures as much as possible that we are doing things with our logical mind and not just our reactive, emotional mind. This allows us to keep to a path of moderation and avoid the excesses that enmities can cause us.