Like stated, an edge is a labelled connection between two resources.
The edge table defines these relations between resources. It does this by adding a directed edge from one rsc record (subject) to another (object). It also adds a reference to the predicate: the label of the edge.
In the admin, edges are represented in the “Page connections” sidebar panel, of the edit page of the subject: the resource where the edges originate. By convention, edges are said to belong to their subject, this to simplify the access control. So if you are allowed to edit the resource, you’re also allowed to edit its outgoing edges (“Page connections” in the admin), creating connections to other resources.
Like categories, these predicates themselves are also again resources: allowing you to specify metadata, give them a meaningful title, et cetera.
Each predicate has a list of valid subject categories and valid object categories (stored in the predicate_category table). This is used to filter the list of predicates in the admin edit page, and also to filter the list of found potential objects when making a connection. On their edit page in the admin interface, you can edit the list of valid subject- and object categories for a predicate.