You can install plugins via the web interface or via the command line.
To install a plugin via the web interface (as you have tried) your webserver must be able to write to the plugins directory (and subdirectories). This is a similar configuration to what is already in place for your “public” directory and config.inc.php’s files_dir. This write permission for the webserver is often appropriately locked down to increase security.
If you do not want to open the plugins directory to be modified by the webserver, you can still install a plugin by copying the files into the plugins directory manually via the command line (or a web front end like CPanel). You should run the upgrade script after installing a new plugin manually.