There are actually three different ways you could have gotten your ASP.NET app over to the cloud, and the confusion comes because you are seeing elements of two of those options.
A Windows Azure Web Site is just a single ASP.NET application running in Microsoft's data centers (versus on premises) so all of the configuration it needs is already part of the project itself (primarily the web.config), some being exposed via the Azure portal. In many ways, this is similar to web hosting providers.
A Windows Azure Cloud Service is a collection of multiple services, which could be web sites/services (Web Roles) or backend code (Worker Roles). The code for your Azure Web Site could also have been deployed exactly as is as a Web Role, but in that case it would be part of a larger application that potentially contains additional Web and Worker Roles. It's the Cloud Service project itself that provides the additional configuration of those web and worker roles (even if you have just a single role in the cloud service). The Cloud Service offering is considered Platform as a Service.
The other option, for completeness, is to create a Virtual Machine image yourself, install your ASP.NET site, and then have Windows Azure host the Virtual Machine. It's at the opposite end of the 'make it simple to deploy' spectrum, but you get a lot of control over what exactly is on the VM that you deploy. This is the Infrastructure as a Service offering from Azure.