I'm not sure I entirely agree with John Am's translation, so I'll have my own go at this. The phrase in question is just this:
ἐπὶ φύσιν νοῦ τε καὶ ἀνοίας ἀφικόμενος
Fowler (translator in the Loeb edition) renders this:
taught him the nature of mind and of lack of mind
"Taught" is a bit more of an interpretation than straight translation for ἀφικόμενος, but that's rather beside the point. The key is that it is the nature (φύσιν) of mind (νοῦς) and the opposite of mind (ἄνοια) that are at stake. ἄνοια, although a word in itself, in Greek seems like a composite of ἀ-νοῦς, the ἀ being privative, similar to how an "atheist" as the opposite of "theist" or how "anaerobic" respiration is opposed to "aerobic" respiration. Hence ἄνοια seems to resemble either a "lack" of "νοῦς" or constitute it opposite.
νοῦς, though it is regularly translated as "mind", has a broader extension than the word "mind" in English, meaning something like reason or even good sense. Hence, ἄνοια, its opposite, has the sense of something that is without reason or foolishness, something like a form of irrationality that does not descend to pure madness (μανία).
In this context, I think what Socrates is trying to ascribe to Pericles is a sort of comprehensive knowledge of the soul (cf. 270b-271c ff. where the art of rhetoric depends on knowledge of the soul). He knows not only νοῦς but the opposite of νοῦς---ἄνοια---and, more importantly, the distinction between them, just as (to extend Plato's analogy in the subsequent passage), the physician must be able to analyse (διελέσθαι) the body, to truly know whether it is healthy or unhealthy and how best to go about restoring or keeping it in health. Now, bearing in mind that Socrates is here assuming that Pericles has become the most perfect orator (ὁ Περικλῆς πάντων τελεώτατος εἰς τὴν ῥητορικὴν γενέσθαι), he seems to be suggest that Pericles, knowing both νοῦς and what is not νοῦς---ἄνοια---has a complete grasp of the soul in all of its modes presentation, both the rational and the irrational.