- You're too small for efficient economies of scale on your own equipment (i.e. AWS is cheaper when considering total cost of ownership).
- You need to scale rapidly to meet demand
The second one is largely a data issue, if you have enough historical data on your customers, their habits, usage, and so on then scaling becomes predictable and even when it isn't you could offload only part of your infrastructure to a cloud vendor.
What's interesting is that several companies that I know which rely on AWS/Azure/et al aren't on it for either of the two above stated "good" reasons.
They are large businesses and do almost no automated scaling. They're on it for what can only be described as internal political limitations, meaning that they are on these services to remove the politics of technology infrastructure, one less manager at the top, a shorter chain of communications, an external party to blame when something does go wrong, and issues like HR/benefits/etc for infrastructure employees is outside the scope.
In effect they view themselves as "not a technology company" so look to employ the fewest technology employees as they can. Even in cases where technology is paramount to their success. It is very interesting to watch, and I'm not even claiming they're "wrong" to handle their infrastructure this way, just that it is hard to quantify the exact reasoning for it.