I believe the Nietzsche's passage referred to is this one:
"Socrates' decadence is suggested not only by the admitted wantonness and anarchy of his instincts, but also by the hypertrophy of the logical faculty and that barbed malice which distinguishes him. Nor should we forget those auditory hallucinations which, as "the daimonion of Socrates," have been interpreted religiously. Everything in him is exaggerated, buffo, a caricature; everything is at the same time concealed, ulterior, subterranean. I seek to comprehend what idiosyncrasy begot that Socratic equation of reason, virtue, and happiness: that most bizarre of all equations which, moreover, is opposed to all the instincts of the earlier Greeks."
Equating reason, virtue, and happiness is the hallmark of moral intellectualism, “one will do what is right or best just as soon as one truly understands what is right or best”. One can find aspects at odds with this position already in Plato's own Chariot Allegory:
"First the charioteer of the human soul drives a pair, and secondly one of the horses is noble and of noble breed, but the other quite the opposite in breed and character. Therefore in our case the driving is necessarily difficult and troublesome."
If the chariot of soul is driven by both reason and passions one may not do "right or best" even if "one truly understands" what that is, for passions may not listen to reason. One need not be a relativist to reject moral intellectualism, it was already seen as hopelessly naive by Nietzsche's time, and dangerous in its naivete at an age of unrest and revolutions, and it was disdained by some contemporary religious moralists. In the context of Christianity an even deeper force than passions opposing reason in the human soul was in play, the original sin, the tainted human nature. Nietzsche himself was particularly influenced by Dostoevsky ("the only psychologist from whom I had something to learn", as he writes in the Twilight), whose novels explored the depths of human depravity, conscious and calculated, in vivid and dramatic terms. And it was even more thoroughly discredited later, after the Nazi and communist terror, in the eyes of Freudians and existentialists alike.
Quite aside from that, Nietzsche was indeed a perspectivist, a position often confused with relativism. Deleuze and Guattari quip about the difference in What is Philosophy?, "perspectivism, or scientific relativism, is never relative to a subject: it constitutes not a relativity of truth but, on the contrary, a truth of the relative." In other words, a relativist believes that there is no truth because things appear differently from different perspectives, a perspectivist believes that each perspective carries a seed of truth. And there is no God's Eye "view from nowhere in particular", as Nagel put it, that can synthesize them all. See Pearson's Perspectivism and Relativism beyond the Postmodern Condition for contemporary discussion, and specifically on how Nietzsche might deal with the usual self-refutation objections to relativism see Nietzsche’s Perspectivism vs Relativism.
Basically, Nietzsche offers his perspective as his own and argues that it is the "right" one for his kind of man at his time, not a universal "truth" for all times. Indeed, the "truth" of his Zarathustra is far from that of the historical Zoroaster, who he nonetheless chose to voice it. Because, as he writes in Ecce Homo:
"Zarathustra was the first to see in the struggle between good and evil the actual wheel in the working of things: the translation of morality into the realm of metaphysics, as force, cause, end-in-itself, is his work. But this question is itself at bottom its own answer. Zarathustra created this most fateful of errors, morality: consequently he must also be the first to recognize it... The self-overcoming of morality through truthfulness, the self-overcoming of the moralist into his opposite – into me – that is what the name Zarathustra means in my mouth."
What was once a great insight into the working of things descended into a fateful error to be overcome, that is truthfulness over "truth", see Nietzsche’s Dance With Zarathustra.