A local storage resource is a reserved directory in the file system of the virtual machine in which an instance of a role is running. Code running in the instance can write to the local storage resource when it needs to write to or read from to a file. For example, a local storage resource can be used to cache data that may need to be accessed again while the service is running in Windows Azure.
References: https://docs.microsoft.com/en-us/azure/vs-azure-tools-configure-roles-for- cloud-service