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Stateless web tier means you don't store any application data in the web server memory or file system. Keeping your web tier stateless enables you to both provide a better customer experience and save money:

• If the web tier is stateless and it sits behind a load balancer, you can quickly respond to changes in application traffic by dynamically adding or removing servers. In the cloud environment, where you pay for server resources only for as long as you actually use them, that ability to respond to changes in demand can translate into huge savings.

• A stateless web tier is architecturally much simpler for scaling out the application. That enables you to respond to scaling needs more quickly, and spend less money on development and testing in the process.

• Cloud servers, like on-premises servers, need to be patched and rebooted occasionally. If the web tier is stateless, rerouting traffic when a server goes down temporarily won't cause errors or unexpected behavior.

Most real-world applications do need to store state for a web session; the main point here is not to store it on the web server. You can store state in other ways, such as on the client in cookies or out of process server-side in ASP.NET session state by using the Redis cache provider. You can store files in
Azure Blob storage instead of the local file system.

Source of Information : Building Cloud Apps With Microsoft Azure