The server does start automatically, as soon as the Windows user account which performed the installation is logging into Windows.
Plex is a “server”. It is not supposed to open a visible window after it is started.
After you have logged on, look into the Windows task tray area. You should see a Plex icon. If it’s there, do a right-click and “Open Plex”.
You can do it like above, or you save a bookmark to https://app.plex.tv to your desktop or your web browser, then use that to open the web app for administration (or the occasional playback, but it is recommended to use one of the dedicated “player” apps for that. Download links are below.)
If you however want to run Plex server as a “system service” (i.e. no matter which user logs on, or even without any user logging in), take a look at PMS as a service
