A subscribing witness is someone who witnesses an event taking place, and signs a legal document verifying that he or she witnessed the event.
Events that often use subscribing witnesses include the signing of legal documents, contracts, etc. Certain legal actions require witnesses to be present while they take place.
The reason why certain legal events require subscribing witnesses is so that the court can have further proof that the event actually took place. For example, a marriage may need a subscribing witness to sign the legal document verifying that the marriage actually took place. The signature from the subscribing witness is further proof that the marriage is legitimate. The subscribing witness can be called upon in the future if there is a legal dispute about the legitimacy of the marriage.