Serious Face is a meme based off of the character Flint Paper from Sam & Max. The meme is used in the sense of "This calls for a serious face!" Similar to Awesome Face and Troll Face meme. The character Flint Paper is known to having an aggresive nature more than a serious one.
The expression is used in a moment where some ridiculous bull crap that nobody cares about is going on, or when something serious or the exact opposite is stated, Flint Paper will give a strict serious face to the reactor or moment.