This use of 'eye' describes the very middle part of something that is usually surrounded by trouble or action. For example, in a storm, it is the calm area in the center.
The eye of the storm is calm.
The eye of the hurricane is dangerous.
In phrases like 'keep an eye on,' this meaning refers to being watchful or attentive to something. It emphasizes careful observation.
Keep an eye on the kids.
He has an eye for detail.
