I don't think we have a lot of internal docs, just high-level explanations like https://databricks.com/blog/2021/05/27/databricks-announces-the-first-feature-store-integrated-with-...
However I don't think there's much to it. The offline storage of a feature store table is simply a Delta table, plus some extra metadata about lineage, managed by a Databricks service. The online feature store data is in a cloud database like Amazon RDS (MySQL) for now. Models trained on features from a feature store log to MLflow with additional data about what features have to be joined at runtime, and these are visible in the logged MLflow artifacts. That's about it for the data architecture.
The feature store does not move featurization code around at all. All featurization code runs in Databricks; the materialized features are stored in Delta and synced to an online store.