Harry meets Ron in the hallway and when they see Slughorn approaching, the professor awkwardly turns around to avoid them. Ron supposes this means Harry still hasn't had any luck getting that memory from him.
Aha! That seems to give Harry an idea: why not use his liquid luck to get Slughorn to talk to him about Tom Riddle?
With Hermione and Ron watching, he drinks the potion and decides to go immediately to Hagrid's. We're not sure why, and neither is Harry, but he's got a good feeling it's the right choice.
On his way, Harry comes upon Professor Slughorn swiping some specimens from the greenhouse. Wait a minute, the professor can't let Harry go roaming Hogwarts grounds at night by himself. So Harry suggests that Slughorn accompany him to Hagrid's.
Slughorn agrees, even though he really just wants to get Harry back to the castle. They come upon Hagrid, who is mourning the death of Aragog (remember the gigantic spider from earlier in the HP series?).
Slughorn tries to be sensitive, but he's also keen to extract some venom from the late Aragog's body. Apparently, it's very valuable in potion making. Hagrid agrees to let him take some.
Then, Slughorn says a few words over Aragog's body, and Hagrid, Harry, and Slughorn go back to Hagrid's.
Once there, the adults seem to get drunk. Speaking of deceased animal friends, Slughorn recollects he once had a pet fish. Kinda makes you think about life…so fleeting…woop, Hagrid passes out in his chair, and Slughorn reveals that his pet fish was actually a gift from a student: Harry's mom.
Slughorn reminisces about Lily. Clearly, he was very fond of her… maybe this will work to Harry's advantage?
Well, not immediately: Slughorn claims that he still can't give Harry exactly what he wants to know, because it will ruin him.
Harry pushes and reveals to Slughorn that he is the "Chosen One"—that is, the only person who can destroy Voldemort.
He uses Slughorn's affection for his mother to his advantage, saying if the professor doesn't share his memory, Lily's death was in vain. Wow, he's going there.
Slughorn finally relents. He begs Harry not to think badly of him when he sees the full memory, and gives it to Harry by pulling it from his head with his wand and placing it in a corked bottle.