Unfortunately, my Japanese ability is nearly non-existent, so I can't provide a translation of the scene on the packaging, but the original Tamagotchi lore can be explained by way of these two Tamagotchi Wiki entries;
https://tamagotchi.fandom.com/wiki/Professor_Banzo
https://tamagotchi.fandom.com/wiki/Mikachu
Basically, the Tamagotchis are tiny aliens who visit other planets seemingly so as to learn more about them, and they were discovered by Professor Banzo and his assistant Mikachu after a UFO carrying some of the aliens crash-landed in the Sumida River in Tokyo.
The creatures couldn't survive in Earth's atmosphere, so the professor created a device that could house them so that a human could care for them; The device resembled an egg and also had a clock function, so it was dubbed the "Tamagotchi House", where "Tamago" comes from the Japanese word for "egg", and "gotch" comes from the English loanword "watch" (the "i" part is silent in Japan). This led to the entire species being named "Tamagotchi", and the original premise of the toy is that you're raising these alien creatures who otherwise can't survive on Earth.
Later generatons of the toys feature the Tamagotchis visiting various locations, and from what I can tell the premise shifted slightly over time, so that you're no longer raising the creature directly within your device, but are instead using it to communicate with a creature who lives on Tamagotchi Planet. (This is further suggested by Tamagotchi: The Movie, where the device belonging to the main human character, Tanpopo, ceases to operate when she ends up on Tamagotchi Planet herself.)
