Hi Divadavid,
It depends on what kind of agency they are. It could be where the agency is actually "employing" you to act as their representatives in their clients' workplace. This would be akin to a contracting company that hires out contractors to their clients but employs the people they are contracting out (if this makes sense). Many recruitment agencies actually do this. You perform the work for the agent's clients and the agency pays you as an employer. In this case, the agency would need all the above details you mentioned.
I suggest that you read the agreement with the agency very carefully. Ask the agency if you have any questions/concerns regarding the actual relationship with the agency and their clients and if you're still unsure why they are requesting this information, you can ask them directly.
They should be explaining all of this to you, in a way that you understand, before you sign with them.