First, he will introduce himself. Not much hidden meaning here.
He then asks you to judge your Pokemon. Dependant on the total number of IVs, he will say these lines:
0-90 Total IVs: "This Pokemon has decent potential overall."
91-120 Total IVs: "This Pokemon has above average potential overall."
121-150 Total IVs: "This Pokemon has relatively superior potential overall."
151-186 Total IVs: "This Pokemon has outstanding potential overall."
So if my Pokemon has a 22/28/20/0/25/28 spread, this adds up to 123, which makes him say the Pokemon has relatively superior potential.
He then proceeds to tell you about your Pokemon's highest IV. He tells you the stat the value is in. If you have one highest IV, he will say:
"Incidentally, I would say its greatest potential lies in (insert best IV here)".
If you have two or more IVs of the same high value, he will additionally say one of these lines in the order of HP, Attack, Defense, Sp. Attack, Sp. Defense and Speed:
"But its Attack stat is good, too."
"Hmm. And its Defense stat is good, too."
"Although its Sp. Atk stat is equally good."
"Its Sp. Def stat seems just as good, though."
"And, well, its Speed stat is good, too."
Then, he tells you the range of values that high is in. Here is what he says:
IV range of 0-15: "It has rather decent stats, I'd say. That's how I judge it."
IV range of 16-25: "It's definitely got some good stats. That's how I judge it."
IV range 26-30: "This Pokemon has some fantastic stats. That's how I judge it."
IV of 31: "Stats like those... They simply can't be beat! That's how I judge it."
Since my Pokemon's highest IVs is a Attack IV of 28, and a Speed IV of 28. He would say:
"Incidentally, I would say its greatest potential lies in Attack. And, well, its Speed stat is good, too. This Pokemon has some fantastic stats. That's how I judge it."
If your Pokemon has any zero IVs, he will also tell you that as well after talking about your highest IVs. He will use the following lines corresponding to the zero IV:
"But its HP... It's pretty dismal, you know?"
"But that Attack stat... It's terrible..."
"But how can you make it through battle with this kind of Defense stat?"
"But this Sp. Atk won't even leave a scratch on an opponent..."
"But this low of a Sp. Def stat is going to leave you high and dry..."
"But you won't get anywhere fast with this low of a Speed stat..."
If you have more than one zero IV, he will also talk about that IV with one of these lines:
"And that's quite a disappointing Attack stat there, isn't it?"
"And that Defense stat is nothing to write home about, you know?"
"And I'm afraid that its Sp. Atk is pretty bad, too..."
"And it looks like it's no great shakes when it comes to Sp. Def either..."
"And that Speed stat... Well, it's nothing to brag about, that's for sure."
My Pokemon also has a zero IV in Sp. Atk. Here is what the judge would say:
"Incidentally, I would say its greatest potential lies in Attack. And, well, its Speed stat is good, too. This Pokemon has some fantastic stats. That's how I judge it. But this Sp. Atk won't even leave a scratch on an opponent..."
The judge also has quotes if you try to judge an egg or a Pokemon is not selected.
"...Oh? You don't need me to judge? I get it."
"Judge an Egg?! That's a tall order even for me!"
All from here.
And some experience attempting to breed Larvestas and getting that 0 IV in Sp. Atk.