(Info for version 2.3.0—other versions may not behave the same way)
You'll receive a letter from Scoutmaster Blake on Campaign Day 21 mentioning heartbroken imps and how to help them. Heartbroken imps will begin spawning on roughly that campaign day, but it's possible that the letter will arrive a few campaign days before or after the heartbroken imps.
Imps can reset (and thus might no longer be there) whenever the game is loaded or the island is regenerated, so if you see a pair of heartbroken imps and want to help them, you should probably do so in this play session rather than exiting to the main menu hoping to resume later to help them—they might not be there when you resume.
Heartbroken imps are supposed to spawn in pairs, but as of version 5.1.0 there's still a longstanding bug that sometimes prevents this from happening. All the more reason to try to help them in the current play session if you see a pair. Also, because of the bug you should probably confirm that the first heartbroken imp you see has a counterpart somewhere on the island before you try to help them.