The phrase indicates there's no single 'front' to the stage; it can be seen from all around. Seeing a concert in the round means you can be seated anywhere and still have a good view.
Pro Tip 2/3
Intimate Setting
'In the round' creates a feeling of intimacy; performers and audience share the same space closely. The poetry reading in the round provided a personal experience.
Pro Tip 3/3
Audience Engagement
Performances are often more engaging and interactive with audiences present in the round. The actors interacted with the audience members sitting in the round.