One of the best writings about Judicial Independence came up at the latest during the prelude to the American Revolution/War of Independence - when the idea was well enough established that the founding fathers were arguing for fully independent legislative powers from judicative courts. On 28 May 1788, in the Federalist No. 78, John Hamilton wrote (emphasis mine):
This simple view of the matter suggests several important consequences. It proves incontestably, that the judiciary is beyond comparison the weakest of the three departments of power; that it can never attack with success either of the other two; and that all possible care is requisite to enable it to defend itself against their attacks. It equally proves, that though individual oppression may now and then proceed from the courts of justice, the general liberty of the people can never be endangered from that quarter; I mean so long as the judiciary remains truly distinct from both the legislature and the Executive. For I agree, that "there is no liberty, if the power of judging be not separated from the legislative and executive powers.'' And it proves, in the last place, that as liberty can have nothing to fear from the judiciary alone, but would have every thing to fear from its union with either of the other departments; that as all the effects of such a union must ensue from a dependence of the former on the latter, notwithstanding a nominal and apparent separation; that as, from the natural feebleness of the judiciary, it is in continual jeopardy of being overpowered, awed, or influenced by its co-ordinate branches; and that as nothing can contribute so much to its firmness and independence as permanency in office, this quality may therefore be justly regarded as an indispensable ingredient in its constitution, and, in a great measure, as the citadel of the public justice and the public security.
The complete independence of the courts of justice is peculiarly essential in a limited Constitution. By a limited Constitution, I understand one which contains certain specified exceptions to the legislative authority; such, for instance, as that it shall pass no bills of attainder, no ex-post-facto laws, and the like. Limitations of this kind can be preserved in practice no other way than through the medium of courts of justice, whose duty it must be to declare all acts contrary to the manifest tenor of the Constitution void. Without this, all the reservations of particular rights or privileges would amount to nothing.
The conjecture "The legislative shall not interpret its laws" is a direct follow up on the idea that judicative, legislative and executive powers have to be as separated as possible, the logic result of the complete independence of the Judicative.
One of the pillars this was founded on is John Locke, who heavily influenced this mindset. He wrote Two Treatises of Government in the late 1600s. In it, he postulated the separation of powers, but he did not propose a judicative power there, as Stanford University will remind you of:
The fact that Locke does not mention the judicial power as a separate power becomes clearer if we distinguish powers from institutions. Powers relate to functions. To have a power means that there is a function (such as making the laws or enforcing the laws) that one may legitimately perform. When Locke says that the legislative is supreme over the executive, he is not saying that parliament is supreme over the king. Locke is simply affirming that “what can give laws to another, must needs be superior to him” (Two Treatises 2.150)
Locke is not opposed to having distinct institutions called courts, but he does not see interpretation as a distinct function or power. For Locke, legislation is primarily about announcing a general rule stipulating what types of actions should receive what types of punishments. The executive power is the power to make the judgments necessary to apply those rules to specific cases and administer force as directed by the rule (Two Treatises 2.88–89). Both of these actions involve interpretation. Locke states that positive laws “are only so far right, as they are founded on the law of nature, by which they are to be regulated and interpreted” (2.12). In other words, the executive must interpret the laws in light of its understanding of natural law. Similarly, legislation involves making the laws of nature more specific and determining how to apply them to particular circumstances (2.135)
However, even under Locke, the legislative still doesn't interpret the laws, that's up to the executive, so one might say, the idea stems from at latest Locke's writing.