The biggest value of Azure that I see is its ease of management, deployment and ability to easily scale thus its ability to supply as much capacity as your applications/users demand.
For example, say you had business application that was seasonal, and in great demand for only a few months of the year, now think of the hassles and costs of not only provisioning this within your own data center, but also ensuring your application could properly leverage additional servers, multi-core CPUs, etc.
As a developer within Azure, you are isolated from these concerns which allows you instead to concentrate on the business requirements and application functionality.
Azure provides a composite set of complex services such as Caching, Identity Management plus a whole lot more that are otherwise complex areas to build, scale and integrate into LOBs, but are instead very easy and prescriptive within Azure. With caching, think session state, think of the complexity of managing session state across multple web servers, with Azure you no longer have to worry about that, its all handled for you as a service, you can choose to leverage (for a low runtime fee) or build yourself.
Queue storage for example, is an excellent way to provide messaging capabilities between decoupled roles. As an example, you could choose to spin up 300 worker role instances to process messages which execute complex business logic and say those messages are coming from 200 web roles which are load balanced across many data centers, but then when the demand goes away, you are able to turn that down to say 3 worker roles handling messages from only 2 web roles. This is a very simple configuration change you can make in a matter of minutes through the Azure dashboard (which as was pointed out in this video, is SilverLight and WCF RIA services). Think of the cost for you if you had to provide that in-house, to not only provision that scenario, but the complexities associated to building that into your applications to take advantage of scale.
Azure Queue, Blob and Table storage are very easy to use and with Content Delivery Network (CDN) turned on, you can push your Blob content close to your target audience with a simple flip of a switch.
SQL Azure makes it easy to migrate and leverage SQL as we do today, except its running on your own VM as part of your composite service offerings.
Azure is not just a datacenter but rather it is composition of services that you can leverage as part of your LOB design (or not) and it is has a number of ever-growing Enterprise features that make it scale to well beyond the needs of most LOB needs.
SilverLight and WCF RIA services are excellent technologies that fit very well into Azure.
I highly recommend that you take some time and complete some of the many Azure SDK as well as WCF RIA Services labs, then revisit this video. I am pretty sure if you do that you
will have many more answers than questions as to why Azure is such an important
platform for developers and the business at large.