This is an interesting quote from the game. I always thought it was some sort of anti-depressant because the guy is clearly unhappy, even during the child years. But if you speak to him as a child wearing the Bunny Hood, Gerudo or Keaton masks he says he's actually a good guy. His aversion seems to be to people, and in particular to his parents, the carpenter being his father. The witch in the potion shop however is his grandmother, and Anju is his sister and he mentions neither of these people as being disgusting. But he goes to the forest for whatever reason, to get this mushroom to make the potion that I guess will help him. As has been said, the blue potion is only available after this so it's fair to assume this is what it's for. Given that he's extremely skinny and unhealthy looking and the blue potion restores health and power, perhaps he really was just unwell. But the quote about there being no cure for a fool suggests that the real problem lies deeper.
When you take care of the pocket cucco for Anju, she says that she'll give you a rare cucco that belonged to her brother and that he'd stopped crowing since her brother had left. This shows that Grog (the punk guy) obviously cared for animals, as has been evidenced already by his liking of the animal masks. Later on Anju also says "my brother must've been very lonely"- I don't know what this could mean, other than another reference to the fact he was unpopular for some reason. Let's connect this back to what his father, the carpenter says, about his son- he behaves ashamed of him saying he just sits around all day and is lazy. If the guy was never seen as good enough for his father, or seemed to receive any other kindnesses- he would probably be extremely unhappy. For all we know, he was mistreated because of this too, as Anju seems to be completely fine, however both share a love of cuccos. When he sees Link has tamed Cojiro he concludes he must be a nice guy and seems shocked at the revelation- and then trusts Link to take the odd mushroom back to the potion lady.
Interestingly, when you get the potion from the old lady, she says it won't work on a monster. As we know from Fado, Grog becomes a stalfos. Perhaps Grog just intended to travel and find out what would make him happy or something- but got physically weak when he reached the Lost Woods, probably because he doesn't have a fairy protecting him from the magic which turns you into a stalfos. So I would still be inclined, therefore, to go with the first conclusion that Grog was depressed and unhappy. But the potion could easily have been for the reason that he was weakened because of his travels into territory he wasn't supposed to enter. All he leaves behind is his father's saw- which shows he might have been intending to do something, maybe even to go to the Forest Temple to stop the terrible things that had been happening- he seemed to have good intentions, whatever he was up to.
