You can think of a countersignature like you might in financial or legal documents (for example, a Power of Attorney):
A countersignature helps confirm the authenticity and appropriateness of what has been submitted.
In DBS terms:
A countersignatory is someone who is registered with the DBS to countersign applications.
They make a formal declaration that:
The role is eligible for the type and level of DBS check requested.
Without this confirmation, the DBS will not process the application.