This is a trope that doesn't appear much in fiction, being pretty much exclusive to Fandom. The basic idea is that the more seriously a character acts, or the more seriously a story takes itself, the more mature it is.
For example, if a Plucky Comic Relief character has a Let's Get Dangerous! moment and spends a good portion of an episode, film, or book being deadly serious, odds are someone will hail this as the character "finally maturing." And when Plucky Comic Relief guy inevitably goes back to making lots of jokes, someone will complain that the writers are regressing the character. While rarely said outright, the implied idea behind this is that having a sense of humor is a sign of immaturity, and that a mature character will act with grim seriousness at all times. "This Is Reality" will often be invoked.
This same attitude can apply to stories as a whole, with comedies frequently seen as being less mature works than their dramatic counterparts, even if the comedy also has some dramatic moments.
Related to True Art Is Angsty and Comedy Ghetto. A leading cause of Darker and Edgier. Fans of this mindset generally want their show to be on the 'Serious' side of the Sliding Scale of Silliness vs. Seriousness all the time, regardless of whether or not it's appropriate, and are often found saying "Silly Rabbit, Idealism Is for Kids!".
A common subversion of the trope is to play it so straight that the characters come off as Comically Serious; the inappropriate or incongruous seriousness becomes laughable in its own right.