>>713022604
He went into the Shelter to try to save Dess, he saw the horrors of the Dark World and tried to tell everyone what happened to her but nobody believed him. He thinks he can "prove" what happened, and if he does that Toriel will love him again and they can be one big happy family again.
Carol "supports" him because he knows the codes, maybe both the Police Code and the Dreemurr Code, and by keeping him under her thumb she has control over access to The Shelter. But he's just her stooge, she only tells him what he needs to know and she's stringing him along with false hope. Toriel resents Carol for emasculating him and enabling his obsession with the case, and Carol is probably using him to spy on Toriel, with him apparently giving her flowers every single day.
The Chapter 5 Dark World will be in the "Flower King" shop, and we'll learn about exactly what happened and what he knows about Dess.