Graphite has some useful characters that make it desirable as an electrode in multiple applications, like electric arc furnace steel production.

The reasons being that graphite is inexpensive, plentiful, durable, stable over a broad range of operating conditions, a good electrical conductor and heat conductor (a lot of heat is generated in electric arc furnace steel production) and it can also expose a "fresh" surface continuously.

The reason why graphite can be a good electrical conductor is because of its free electrons. Since the graphite is made up of various Graphene layers, it can serve as a good host material for ion storage. So there are broken bonds that can free electron and meanwhile increase conductivity. So graphite is conducting in nature and therefore is used to make electrodes.