The story of this cat is probably one on the best known physics paradox of all time. Schrödinger’s Cat is a thought experiment trying to give a concret exemple of how would apply the Copenhagen interpretation of the quantum mechanics to larger scale object. The Copenhagen interpretation defends that we can not know the state of two quantum particles until measured and both state are possible at the same time.
In this experiment a cat is placed in a box with a bottle of bottle of poison and a small amount of a radioactive substance, in one of the two state of the radioactive substance the bottle would fall and the cat would die and in the other state nothing happens. Since we can not know the result before measuring it, the cat is dead and alive at the same time before opening the box. So how does it became so popular ? It still talking about quantique physics which is probably the less fun topic of all time.
Credits: Kevin Frank
I think that the reason why it became so popular is that for a lot of people, quantum physics is the ultimate stade of the geek. There's no further point. For instance four references of the Schrödinger cat are made during the Big Bang Theory show. For exemple in "The Russian Rocket Reaction" Sheldon describes is relationship with Leonard as "Schrödinger's friendship" because he likes him and hates him at the same time.
An other reason might be the simple fact that it's about a cat and cats are the icon of the internet. If Schrödinger had a snake would really someone care about this story. No it's all about the cat.
The mythos of the Schrödinger cat has started long before internet and a lot of author have choose to talk about this topic in their books, and the real phenomena started when the internet, movies, comics and other TV shows took the story to really create out of it a running popular yet a bit nerd gag.
Futurama Season 6 Episode 17
Erwin has to explain if the cat is dead or alive.
The famous cat has even find a place in video games, films and card games here are two exemples.
Card from Force Of The Will
Card from the amazing exploding kittens
Here is a great video to conclude this post.
There is still so much more to say about this famous cat, here is a great article to go further into that topic: https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture#Literature