The first races to discover the Khajiit were believed to either be ancient Nord descendants or Bosmer (wood elves ) explorers who happened to stumble into the lands of Elsweyr. Various poems and stories centered around Tamriel have usually mentioned things about "A strange feline race that lives amongst the desert" or "Cat people that watch the moon". When these tales were written, it was widely assumed that the Khatjiit came from one of the moons above Nirn. Of course, these stories are very inaccurate or incomplete, so it is best not to trust them.
Khaj means "desert" and iit means "where on lives". So, Khathiit essentially means "Desert Liver". Khatjiit language is difficult to decipher for other races, its very simple yet takes time to grasp how to fully speak a sentence of the language (often resulting in complete gibberish most people will find insulting). Many scholars agree that Elsweyr naming is similar to Black Marsh native naming (Argonians have a separated first name, such as Shady leaves-swimmer, and so do Khatjiit. Despite this, the citizens of Black Marsh rarely interact with those of Elsweyr.
As you may know, Khatjiit culture is largely centered around the moon. In fact, the two moons are centered around Khatjiit birth. With over twenty species of Khatjiit, the moon and its lunar cycle determine what type of Khathiit will be born. Most Khatjiit are usually very small at birth, about an inch taller then a 3 week old kitten. Some are kept as pets at birth by other Khatjiit and then released once they became about 5 years old. Something interesting to note is that many of the things they worship often have something to ddo with the moon.
This is just a small fraction of the history, culture, and biology of the amazing felines of Elsweyr!