Conor's a dude with some seriously scary nightmares, not to mention way too many (two too many, to be exact) monsters for one person to handle. Somehow, though, the kiddo manages to flit between the monsters' realities and his own reality and he takes us with him when he goes. Throughout A Monster Calls, the real world and the monster world inform each other. We can understand what goes down in monster land because we understand what's going on in Conor's life and vice versa.
Conor subconsciously summons the monster because he needs somebody to be honest with him, just like he needs to be honest with himself.
The remnants of the monster that come back to the real world with Conor after its visits are similar to the ectoplasm that supposedly covers psychics after séances. In other words, they're total, irrefutable proof that the monster is real.