Technically it allows for constrained Delaunay triangulation.
In simpler terms, it allows you to enforce that there is an edge between 2 vertices. This affects how the mesh triangles are created, which affects how the mesh deforms.
See this part of the Spine User Guide, including the section below it that has example images:
http://esotericsoftware.com/spine-meshes#Edges
Often features are surrounded, like the mouth or eyes in your image. That "protects" the feature from unwanted deformation. When triangles (like the cheek) outside the orange edges stretch, the eyes won't stretch.
Additionally, the orange edges can be dragged, allowing you to more easily select or move vertices. shift+click does edge loop selection which is a very fast way to select many edges in a line. For example, the entire eye orange lines can be selected with one click.
