There are two aspects to the question, “why do cheetahs have spots?”. The first is why they exist and the answer is because they provide camouflage for the animal so that he/she can be more effective when hunting.
The second aspect of this question is how the spots are formed. That is a far more complicated question. You will notice that there is a similarity between a spotted tabby domestic cat and a cheetah. In fact, there are many wild cats with what appear to be tabby-like coats and variations upon the tabby coat.
In the past, I have suggested that the same genetics are involved in creating the domestic cat’s tabby coat and the spots of a wild cat but this may not be the case. Although a scientist at Stanford University, Christopher Kaelin, says that the same genes that produce the cheetah’s spots also control the house cat’s coat. Notwithstanding this uncertainty, the process is very similar and the reasons for their existence is exactly the same.
So there’s not much difference between tabby cats and cheetahs in respect of their spotted coat. It has been suggested that the pattern creating process is shared by all cat species and that the design and style of the pattern is just due to tweaking by the genes when the cat is developing as an embryo.
Cats, whether they be wild or domestic, are born with their particular coat pattern. When the process of developing the coat pattern is completed it is fixed so as the cat develops and grows up the pattern simply becomes larger and the markings expand but the same pattern exists.
The Science Illustrated website says that the loss of one of the genes called ‘Taqpep’, “encodes a protease1 released from the surface of cells and disrupts these colour patterns without affecting any other part of the cat in question…a diffusible protease helps to organise skin patterns during development.”
I think it fair to say that humans do not yet fully understand the workings of the genes which dictate the creation of the spotted tabby-like coat in cheetahs. Suffice to say that the answer the question “why do cheetahs have spots?” is because they carry certain genes which dictate how the coat colour and pattern is developed during the early years of the cat’s life.
Ref: “Protease” means: enzyme that performs proteolysis.