I'll answer based on the Hellboy comics. I don't know if the first two Hellboy films are regarded as canon, but I have never viewed them as such as they seem to have little connection to the comics. The 2019 film is very close to the comics, though sadly that didn't work out too well.
Anyhow it is a recurring theme in the comics, and especially in the sequel Hellboy in Hell, that Hellboy is destined to rule Hell and deliver all sorts of vaguely defined mayhem. He rejects this destiny, and snapping off his horns is a symbol of that rejection.
The first time we encounter the horn snapping is in Wake the Devil part 5 when Hecate specifically invokes his destiny and sends him down to Hell:
We see the horns grow as he descends in Hell to take up his destiny:
And he explicitly rejects that destiny and snaps off the horns as a symbol of that rejection:
This theme recurs again and again in the comics. I'm not going to post scans of every occasion otherwise this would be a long answer, to say nothing of the copyright issue, but it's the overarching theme of the comics that Hellboy was born of a mortal woman and the archdemon, Azzael, and is destined to rule Hell. This comes to fruition in the final episode of Hellboy in Hell, though I won't say how for fear of spoilers :-)
In the 2019 film Hellboy snaps off his horns to reject Nimue's attempts to persuade him to take up his destiny. This doesn't happen in the comics because the battle with Nimue ends rather differently, but again I won't give spoilers.
So the bottom line is that Hellboy snaps his horns as a symbol of his desire to live a normal life as a mortal among humans. I don't think we ever see Hellboy filing his horns in the comic. Whenever he snaps the horns they leave a smooth surface, and I don't recall any suggestion in the comics that the horns grow naturally like hair or nails do.