The architecture of a native app in Snowflake involves creating an app package which serves as an installer containing necessary data and user-defined functions (UDFs). The app package includes a manifest file defining metadata, artifacts required, and access permissions. Users can create Python UDFs using Anaconda for package management and leverage a variety of packages in their native app. Native apps can be shared on the Snowflake marketplace or within an organization, encapsulating specific functionalities for others to install and utilize.