There's a lot that goes on behind the scenes when you deploy an application to the Azure cloud. I don't have any special insight into what's going on behind the curtain, but having worked on the VS tools to upload projects to the Azure cloud, these are my impressions as an outsider looking in:
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Among other things:
1.Hardware must be allocated from the available pool of servers
2. The VHD of the core OS must be uploaded to the machine
3. A VM instance must be initialized and booted off that VHD image
4. Your application package must be copied to the VM and installed
5. The VM monitor must wait for your service to start up, or fail
6. The data center load balancer and firewall must be made aware of your application's service endpoints
7. Once all of that has synchronized, your app is accessible from the web.
The VHD image is probably gigabytes in size, much larger than your app upload. Even on a superfast datacenter network, it takes time to move that much stuff into the VM, unpack it, and boot from it. Also, the load balancer and firewall are probably optimized to make routing requests the highest priority. Reconfiguring the firewall and load balancer is a lower priority, and has to be done without interrupting traffic flow.
Also, note that all this work only has to be done for a new deployment. Updating an existing deployment rolls out much faster - 2 to 3 minutes instead of 20 to 30 minutes.