A zarpe is a port clearance document. In the us it's form number 1300 (http://forms.cbp.gov/pdf/CBP_Form_1300.pdf
) Essentially its a piece of paper that tells the next port authority that you were authorized to leave your last port legally.
As far as the whole anchoring fees thing, you need not worry about things like that. Do your due diligence, arrive at a port of entry, and enjoy yourself.
FYI, getting the 1300 form filled out in the US is interesting sometimes. Once outside of the US though you must clear out of a country before moving on and a form similar to this is just a part of the package.