I don't disagree with the fine answer posted by Technik Empire. This is more in the vein of an extended comment/elaboration upon it.
Public lands, especially those suitable for supporting livestock, were indeed referred to as 'the common', so called because they were 'held in common', owned by all members of the community, rather than by any individual. Many public spaces retain the name today, as with the Boston Common, the oldest public park in the US.
Public lands of this sort also lend their name to the Tragedy of the Commons, which describes the incentive of members of a community to consume public resources - originally, this public grazing land ('the commons'). To combat this tendency, many communities made laws limiting who might use these public lands for their animals, what animals they could place there, and so on.
It is in this sense, then, that the subject of the quote is running his illegal beasts. 'Run' here is used in the sense of 'herding' - the accusation is that the subject is deliberately encouraging his animals to graze on public lands (thus 'despoiling' them) in a manner that is against the law.