Plato defined man as a featherless biped [Statesman 266e], and was much applauded [for that]. [Diogenes heard this and so] he took a plucked chicken into the school and said, "here is Plato's man." So Plato's definition was changed with the addition "with broad nails."
Get it? I didn't think so. Let's try it one more time in Greek:
Πλάτωνος ὁρισαμένου, Ἄνθρωπός ἐστι ζῷον δίπουν ἄπτερον, καὶ εὐδοκιμοῦντος, τίλας ἀλεκτρυόνα εἰσήνεγκεν αὐτὸν εἰς τὴν σχολὴν καί φησιν, "οὗτός ἐστιν ὁ Πλάτωνος ἄνθρωπος." ὅθεν τῷ ὅρῳ προσετέθη τὸ πλατυώνυχον.
Still don't get it? Well, it's not really a very good joke, but it's a joke. And perhaps it's worth ruining it with an explanation.
The introduction is classic: Diogenes walks into a bar and hears that Plato defined a chicken as a featherless biped. The detail about the applause is evidently intended to amuse us (who could applaud such a ridiculous definition, ridiculous even within the dialogue, as the Stranger notes in the Statesman?), not to report a fact. The punch line of the joke is that τὸ πλατυώνυχον “the quality of having broad nails” sounds in Greek much like τὸ πλατώνικον, “the platonic thing;” the “improved” academic definition is thus “man is a ζῷον δίπουν ἄπτερον πλατυώνυχον/ πλατώνικον,” a “‘platonic’ featherless biped.”
Ok, I said it wasn't a great joke, but I think it's nice to know that people in antiquity took that part of the dialogue with a sense of humor.