Hey Aindreas,
Events will remain visible on the front-end even if they have passed – this is true regardless of the calendar view. For example, if today is August 16th and you view the “Month View” grid calendar of events on your site, the dates for August 1, 2, 3, etc. will still show up, and any events from these days will be there too.
It IS possible to hide events after they pass, though it would take some custom code a bit beyond the scope of support forums here.
Aside from all of this, however, it’s important to point out that the link you provided is not a an Events Calendar view – that link is just a “category” archive of “normal” Posts on your site, not Events. These are normal posts where it seems you’re trying to “embed” event data with a shortcode called [event], so this is not something we can help with much since it’s just your Posts archive on your site, and not something generated by The Events Calendar itself.
I hope this information helps clarify some things – let me know if that’s the case or any other thoughts you might have on this.
Cheers!
George