Now, first off, my studies never dove into big cats, so I can only speak to methods of striping generally. I would suspect that a mechanism like I will explain is responsible, but other solutions may exist.
The OP seems to be asking the WHY while acknowledging that good explanations of the HOW exist. This seems odd as good reasons for the why are much clearer. Coloration patterns are often reinforced through camouflage and/or mate selection (again, examples, other factors could also reinforce these changes). The best answer here seems to be camouflage. As far as the OP, the answer can end there as he doesn't seem to take an issue with the how.
There is also an implied question that does get at the how though. The OP seems to suggest that the hair coloration is controlled by a single gene per hair, requiring one mutation per hair that changes color. This is wrong and gets at a misunderstanding of the how. Now, as I said, I never studied much in the way of big cats, but other animals have stripes too. Insects, for example, often develop stripes through pair rule genes and turing reaction/diffusion models. Now, I was looking for a good approachable graphic to help explain these, but was unable to immediately find anything. I'm thus going off my remembrance of lessons from a decade ago. Mistakes may be present.
Basically, multiple genes are at play that result in kind of standing wave patterns. Let's say you have some gene X that's expressed at the head side, and forms a gradient towards the tail:
XXXXXXXxxxxxxxxx------------
Now, say a second gene, Y, is also produced
XXXXXXXxxxxxxxxx------------
yyyyyyyyyyyyyyyyyyyyyyyyyyyy
Now, here's where it gets interesting. X promotes X and Y, but Y inhibits X. Let's go through a bit slower. Working left to right.
XXXXXXXXXXXXXXXXXXXXXXXXXXXX
yYYYYYYYYYYYYYYYYYYYYYYYYYYY
X promotes Y and X
Xx--------------------------
yYYYYYYYYYYYYYYYYYYYYYYYYYYY
which suppresses X
Xx--------------------------
yYYy------------------------
which leads to Y tapering off
Xx---xxXXXXXXXXXXXXXXXXXXXXX
yYYy------------------------
letting X get expressed again
Xx---xxXXXXXXXXXXXXXXXXXXXXX
yYYy---yyYYYYYYYYYYYYYYYYYYY
Which promotes Y
and so on until you get
Xx---xxXXx---xxXXx---xxXXx---xxX
yYYy---yyYYy---yyYYy---yyYYy---y
Have either of those get linked to genes for hair/skin coloration, and boom, instant stripes.