Kameko and Grand Bois Cheri are the pets of Toko Fukawa and Celestia Ludenberg, respectively. Kameko is a female stinkbug belonging to Toko. Grand Bois Cheri is a male cat belonging to Celeste.
There is not much known about these characters, as they only appear as hostages in Danganronpa Another Episode. We can assume that they were the only people featured in Toko and Celeste's motive videos in chapter 1 of Trigger Happy Havoc. Although they have not appeared in any Danganronpa game, they may have more in common than meets the eye.
Kameko and Grand Bois Cheri are both incredibly loyal. Kameko was the only one to tolerate Toko and her stench (although a good amount of her smell is actually caused by the stinkbug.) Grand Bois Cheri was Celeste's only real friend, and one of the only beings that Celeste tolerated. Not being humans, as well as being very close to members of the killing game, these two characters would likely get along well.
Fans of the ship mostly like it because they are both pets of main Danganronpa characters, as explained above. Since they do not have speaking roles in the games, not much is known about their personalities besides what we learn from their hostage bios in UDG. This caused fans of the ship to headcanon their personalities and roles as well.
Grand Bois Cheri, being the man, is normally portrayed as the more dominant one. As he is very close to Celeste, it is likely that he would behave like her. The same thing goes for Kameko acting similarly to Toko. Grand Bois Cheri is portrayed as snotty, reserved, and luxurious, and Kameko is usually portrayed as clingy, self-deprecating, etc.
Fans of this ship usually also ship their owners, Celeste and Toko, together. Fan content often branches off of this ship, like if their owners went on a date and they were taken along, they fall in love. A popular trope for the two is "forbidden love" because their species don't typically mix, and in canon their owners are not on incredible terms.