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From https://curl.haxx.se/docs/httpscripting.html

In the HTTPS world, you use certificates to validate that you are the
one you claim to be, as an addition to normal passwords. Curl supports
client- side certificates. All certificates are locked with a pass
phrase, which you need to enter before the certificate can be used by
curl. The pass phrase can be specified on the command line or if not,
entered interactively when curl queries for it. Use a certificate with
curl on a HTTPS server like:

curl --cert mycert.pem https://secure.example.com

curl also tries to verify that the server is who it claims to be, by
verifying the server's certificate against a locally stored CA cert
bundle. Failing the verification will cause curl to deny the
connection. You must then use --insecure (-k) in case you want to tell
curl to ignore that the server can't be verified. More about server
certificate verification and ca cert bundles can be read in the
SSLCERTS document, available online here:
https://curl.haxx.se/docs/sslcerts.html

At times you may end up with your own CA cert store and then you can
tell curl to use that to verify the server's certificate:

 curl --cacert ca-bundle.pem https://example.com/

What HTTP request and response headers do curl --cert and curl --cacert read and write?

Thanks.

Note that TLS is independent of HTTP. Other protocols may be passed over TLS (eg LDAP, SMTP, NNTP,...). Basically, TLS does "low level encryption" of the data channel, and then you can pass anything over it.

With HTTPS, the HTTP traffic is layered on top of TLS. Things that happen at the TLS layer (TLS version; cipher negotiation; certificate validation; SNI) are "low level" compared to HTTP, and invisible to the HTTP layer.

Telling curl to use --cacert and --cert will tell the program to send a client side cert to the server, which is processed at the TLS layer, and is invisble to the HTTP layer. A web server may expose the CN of the client to a CGI program, or expose it to an authentication module, but this is all done outside of the HTTP request.