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To containerize an ASP.NET Core application and SQL Server, developers need to create a Dockerfile for the Web API project, change the connection string to point to the SQL Server container, build the Docker image, and define a Docker Compose file to start both containers. The Dockerfile performs a multi-stage build, pulling down two images: one with the .NET SDK and another with the ASP.NET runtime. The connection string is updated to use the SQL Server container's name instead of localhost. A Docker Compose file defines two services, one for the Web API application and one for the SQL Server container, mapping ports and setting environment variables. Finally, the containers are started using `docker-compose up`, allowing developers to run the application in a Docker container with the SQL Server database.