The name Beelzebul was used in Jewish discourse as a designation for the chief of demons or Satan. The form is related to the Old Testament name Baal-zebub associated with the god of Ekron, although the precise history of the later form is debated. Jesus' listeners would have understood the accusation as extremely serious: His opponents were attributing works of deliverance to demonic power.
The image of the strong man comes from the familiar problem of household security. A robber could not freely plunder the property of a powerful householder unless the householder were first restrained. Jesus turns that ordinary image into a picture of spiritual conflict. The liberation of people from demonic oppression is evidence not of cooperation with Satan but of Satan's power being challenged and overcome.