First off, remember that cloud computing doesn't fit all circumstances. There are forces for and forces against, and these vary from company to company according to the details of what they're doing. Also remember that there is a continuum between cloud computing and traditional hosting.
For a small or medium business, the big gain of cloud computing is that they can gain access to server-based computing without having to own all that physical infrastructure related to a properly set up machine room. This can be a gigantic saving; machine rooms are a big investment when done properly. Moreover, a cloud provider typically has much better network access than most small businesses. SaaS makes a lot of sense too at this scale, since it may well allow a small business to get away without hiring a full-time system administrator; at the small end of the scale, that's a substantial saving.
For a large business, the gain is more to do with being able to have more flexibility so that they can take up opportunities rapidly. They can also use the fact that there are many providers who are geographically distributed to reduce the risk of outage killing their customer-facing presence. (Not all do; resilience costs money.)
In all cases, the big wins have to do with not having to own a building that houses a server room, not having to maintain and insure that building and server room, possibly not having to purchase and maintain the hardware in the server room, etc. Moreover, a cloud provider is a specialist in hosting and so is well-motivated to focus on reducing costs through measures like using efficient solutions for power and cooling; most organizations – even large ones – won't have a lot of expertise in that area.
So in short, the win of cloud computing is the regular win of outsourcing supply of a service that is not a profit-center. Companies do that all the time (e.g., they probably outsource the cleaning of the offices too, and virtually everyone outsources non-emergency power generation if they can). Don't view cloud computing as different, view it as part of the same thing.