Astronomer is a data company utilizing metrics to track the growth of Apache Airflow by analyzing development activity, demand, and community population. They monitor development through GitHub data, focusing on commits, pull requests, and issues, and analyze conversations and votes from Airflow mailing lists. Demand for Airflow is gauged through download metrics from PyPI and Docker Hub, indicating a vibrant and growing interest in the platform. Community growth is assessed by tracking individual contributions to the codebase and interactions on mailing lists, alongside monitoring Slack, which hosts over 25,000 registered users. Data is gathered and managed using Airflow DAGs, and Astronomer offers a free ebook to help beginners get started with Apache Airflow.