The site you are seeing on the web role and inside the cspkg file is the output of the compile, so you can't get the original .cs files out of them. That said, you can use a tool like Reflector, Just Decompile, or a variety of other decompilers out there to reverse engineer your compiled bits into something that will be very close to the original C# code (not I'm assuming this is your own code, or code that doesn't have a provision against reverse engineering). This at least will let you use the bits on the webrole to get the majority of your code back, then review it to see how good a job it did.
Note, you can open the cspkg file. It's just a zip file. You can rename with a .zip file extension and open it up, but you won't find the .cs files in there. The only time you find this to be the case is if you have multiple websites within a single web role. The default packager for Windows Azure doesn't compile the additional sites, only packages up all the files in their root directory. Not at all helpful for actual deployments really, but this won't likely help you.
You are likely well ahead of me on this, but I'd recommend using a personal source control system of some sort to avoid this issue in the future.