(see also: Paths Overview)
A self-Intersecting path is a single path that intersects with itself. These self-intersections can sometimes have odd relationships with how lines and shapes are created. These interactions only occur on Closed Paths.
Self-Intersections
For example, below is a single path with a few self intersections. It intersects multiple times, but the intersections are not compound. On the right, you can see the kind of shape it creates when filled. It looks like 4 separate rectangles, but in reality they are one shape.
This can either cause problems for your design or you can learn to leverage it to create cool effects. If it's causing a problem, you can remove Self-Intersections and split the path into shapes. Go to Arrange > Path Direction > Automatic. Each intersection is now its own object. Because a self-intersecting path would confuse the path direction, the program is forced to split it, hence this functionality.
Compound Self-Intersections
A compound self-intersection occurs when more than one self-intersection occurs inside a single element of the path. This forces Flexi to make a decision about which portions of the object are filled and which are not filled. You can change his decision in the Fill/Stroke Editor.
(see: Even-Odd Fill/Winding Fill in Fill/Stroke Editor - Solid Fill)
No Fill Even-Odd Fill Winding Fill
Again, you can use the Automatic Path Direction to split the self-intersections into separate shapes. Go to Arrange > Path Direction > Automatic. Regardless of the fill type you have selected, it will always treat it as an Even-Odd Fill when splitting the parts.