It’s a legal term for handling data. It’s when Anthropic uses an external party to handle their data / systems, but Anthropic is the legal entity responsible for the data privacy, as the customer (you) has a contract with Anthropic.

https://www.gdprsummary.com/gdpr-definitions/sub-processor/ in context of GDPR/DORA it is about formalized agreements with 3rd parties that will handle data, your own and of your clients.