see weewx util/systemd/weewx.service for an example config file. you will copy this to /etc/systemd/system/.
see online manual systemd.unit(5) to understand its content (you are unlikely to need to change the file)
see online manual systemctl(1) for the commands to start/stop weewx service, once configured as above.
the key command are
# systemctl [start | stop | enable | status] weewx
(after copying file into /etc/systemd/…, you should run
# systemctl daemon-reload
to ensure systemd realises it has been updated)