Most of the curl web site is also available in a public git repository, although separate from the source code repository since it generally is not interesting to the same people and we can maintain a different list of people that have push rights, etc.
The web site git repository is available on Github at this URL: https://github.com/curl/curl-www and you can clone a copy of the web code like this:
git clone https://github.com/curl/curl-www.git
The web site is a custom-made setup that mostly builds static HTML files from a set of source files. The sources files are preprocessed with what is basically a souped-up C preprocessor called fcpp and a set of perl scripts. The man pages get converted to HTML with roffit. Make sure fcpp, perl, roffit, make and curl are all in your $PATH.
Once you have cloned the git repository the first time, invoke
sh bootstrap.sh once to get a symlink and some some initial local files setup, and then you can build the web site locally by invoking
make in the source root tree.
Note that this does not make you a complete web site mirror, as some scripts and files are only available on the real actual site, but should give you enough to let you view most HTML pages locally.