In the phrase “Regibus exterminatis, libertas in re publica constituta est”, which type of phrase construction is “regibus exterminatis”?

First of all, notice that the phrase is divided in two parts. The first part contains a name in ablative case, acting as subject, followed by a past participle acting as verb. The second part forms an independent sentence with no grammatical connection to the first part.
Therefore, we came across an Ablative Absolute construction (from latin absolūtus, free from connection) that must be translated as a subordinate. Finally we can translate the sentence: “having expelled the kings, freedom was established in the state”.