To anyone still interested in this, I'd say Conifold's extract from the SEP and Dyske's essay are great starts, but I've found the clearest exposition to be in Luc Ferry's Political Philosophy (Volume 1). There he compares natural right (ideal) to positive right (real).
He explains that Strauss was against the Enlightenment philosophy which led to modernity. Strauss characterizes this through his "three waves of modernity" which explains the process by which modern philosophy grew to seek the coinciding of the ideal and the real. This is opposed to classical philosophy, where the real and the ideal are separate and distinct. For Strauss's classical philosophy, the ideal = the cosmos -- the order and rationality inherent in nature. In this sense, Natural Right is not "natural" in the way we use the word. It is referring to the order of the cosmos, which we (the Real) have to work toward in order for our souls to be harmonious with it (the Ideal).
Machiavelli, the first wave, sought to "lower the standard" of the ideal, so that instead of a hard-to-achieve rational cosmic order, that relied very much on chance for its actualization, we could replace the old ideal with a new one. Most of the populace cares primarily about peace, health, and prosperity according to Machiavelli (see The Prince). So, let's make that the new ideal (achieved through technology - the burgeoning natural sciences at the time). In this way, the ideal becomes accessible through the real and much more attainable.
The ultimate culmination of this was German Idealist philosophies of history (the third wave of modernity), such as Hegel's, which posited that the progress of history constituted rationality itself. Progress - what is Right - is achieved through the "cunning of reason" (passion) operating through history. This is the ultimate fusion of the real and the ideal, where there is no distinction between them at all. And this is what Strauss was arguing against.
So Natural Right = based in a distinct ideal, and Positive Right/Political Right = basing the Ideal in the Real (Real = that which exists positively, think positivism).
This was written in a rush and is no polished exposition or philosophical tract, but it is the gist of what I've gathered.