Fuck your mother
No, no! Please, let me explain!
In Mandarin the phrase "grass mud horse" (草泥马, cao3 ni2 ma3) is a play on the phrase "fuck your mother" (肏你妈, cao3 ni3 ma1).
Let's back up for a second. In Mandarin, there's roughly 300 possible syllables with possible 4 tones each (well, 5 if you count the neutral tone, but almost no one does.) Those possible monosyllables have to verbally represent the over 10,000 existing characters. At this point, you might wonder how the hell you can even speak the language with so much ambiguity. It might seem like a problem, but context is a huge help.
That said, the language is ripe for wordplay. And nothing seems to get people in modern China stoked to do a little wordplay and punning than government prohibitions against anything, especially smut or vulgarity.
So, if you say "grass mud horse" in Mandarin, it sounds strikingly similar to saying "Fuck your mom." In verbal communications you'd never do this, but in written communications it creates a delightful ambiguity.
Someone, at some point, came up with the grass mud horse as a fictional animal. Whose name also happens to sounds like telling someone to fuck their mother. When censors crack down on telling people to fuck their mother, or when social norms demand you not say it, the grass mud horse rears its adorable quasi-alpaca head. Frankly, when you can find a plush doll for the grass mud horse, a whole song, a video, etc. you know it has developed into a full grown meme. I assure you, it's Big on the Internet.