I think that means that if you want to operate as if you are a charity then you must register with the CC. There is no obligation for a church to operate as a charity. They could become a non-charitable club and lose the benefits of charitable status.
I understand the law changed a few years ago - when suddenly each congregation became its own charity.
The public remit would be the public meetings they hold and the public ministry they conduct.
I think one of the things is having a public sign showing meeting times on the place of worship - I believe that may be an issue for the (Exclusive) Plymouth Brethren