Handling Webhooks with Java, Spring Cloud Function and Azure Functions involves setting up a serverless function to respond to incoming messages from Twilio's API. The function needs to be deployed to an Azure Functions platform, which provides a public URL that can handle incoming requests. The process involves creating a new Spring project using the Spring Initializr tool, defining a WebhookHandler class that handles the incoming request, and deploying the function to Azure using the Azure Functions Core Tools. Once deployed, the function can be tested with curl commands or an SMS test, and it can also be configured to use Twilio's API for sending responses to incoming messages. The function can be made more useful by adding dependencies and configuring the Maven configuration file. Finally, serverless cold starts are discussed, and the Azure Function Premium Plan offers guaranteed-warm function invocations for a price.