Capistrano should be run from your workstation, not from the server. It is designed to deploy the software to the server, and the server's own configuration (monit, upstart, init.d scripts, launchd, dbus, etc) are responsible for running the daemons.
Capistrano includes a legacy "start" command from the days before Rails had Passenger mod_rails and one had to explicitly start a mongrel pool on every deployment, this hasn't been true for a number of years. The 'restart' command exists for anyone not in a position to use monit (or similar) to restart their daemons in a sane way, this applies to users of Unicorn (and thus Rainbows & related) where monit can't/won't signal URS2 for a seamless restart, and users of mod_rails, who can achieve better performance from touching ./tmp/restart.txt, rather than restarting the whole apache daemon.
The "stop" command also exists, in the rare case that someone really wants to stop their production daemons, again - this made more sense in the days of script/spin (obsolete since 5 years?) - "stop" is more commonly replaced by "web:disable" - replacing the index site with a static HTML file for deployment safety reasons.
I hope this clarifies the purpose of the tool for you.