How can I fix this? I tried downloading curl but I have no idea how to install or use it or what it is even for.
Curl is basic requirement of many plug-ins.
It is for making request server to sever. Or in this case talking to google recpacha to verify.
If you don't know how to install it, ask your host. You need curl, php-curl and the following functions to be enabled curl_init, curl_setopt, curl_exec. I have seen instances where curl is installed but the host has disable curl functions.
grep is your friend.