First, imagine having a database that allows one to freely create record types. One might have standard data types like contacts, events, emails, checks, expense reports, etc.
These record types are nodes. Now imagine being able to connect these nodes using any type of edge you like. For example a contact might be connected to an event as an "invitee". Thats how the edge would be labeled. Now the relational folks will say that that is a relationship that could be predicted. But at some point, some new type of record is created. And you as a user want to connect that record to existing records. For example you have added a "shoe" record type to keep track of all of your shoes. You then decide you want shoes to be connected to events so that you can map what shoes you wore to what events. You don't want to modify your schema. You don't want to add a new mapping table, you just want to connect the record. And you want to be able to query the graph for all the things of any type that are connected to that record. More importantly, you want the end user to be able to decide that it would be useful to connect shoes to events since no self respecting programmer is ever going to design such a system.
This is the type of flexibility that you need in a web application that will evolve over time. But the minute you want to connect that new record type to the existing object, you either have to modify your schema, or you have created a database that is highly flexible via totally generalized mapping tables, but is not optimized for these kinds of structures. For example just creating a giant mapping table to connect objects will work in an RDBMS but it is not at all optimized and will fall over at scale. Since we are building something that will handle awesome scale, using an RDBMS in this way was a non-starter. Philosophically, we probably have more in common with Google BigTable than with an RDBMS.