Three legal pictures run through the Western accounts of the atonement, and telling them apart is the quickest way into the governmental theory.
In the first, the picture is private debt. Something is owed to God, humanity cannot pay it, and Christ supplies the payment. That is Anselm.
In the second, the picture is a criminal trial. A law has been broken, a sentence attaches, and the sentence is executed on a substitute. That is penal substitution.
In the third, the picture is public administration. God is neither a creditor nor exactly a judge but a ruler responsible for an order, and the question is what a ruler can do without destroying the order in his charge. That is Grotius, and he was a working jurist as well as a theologian, which is visible on every page.
The move that follows is the one everything turns on. A ruler is not obliged to exact the exact penalty the law names, because the law exists for the sake of the order rather than the other way about, so the penalty may be relaxed. What the ruler cannot do is relax it silently, since that would teach the governed that the law does not mean what it says. So the relaxation is made public and costly. The death of Christ is a penal example: it displays what the law holds sin to deserve, and having displayed it, God may forgive without the law being emptied.
Two consequences are worth stating plainly, because they mark the theory off from its neighbours. On this account Christ is not punished in the sinner's place, and the exact penalty the sinner incurred is never borne by anyone. What is provided is an equivalent sufficient to uphold the order. That is why the governmental theory can be described as objective, since something real happens outside the human being, while still denying the transfer of penalty that penal substitution requires.