You need a whereHas with a whereIn statement. The code would be something like this

$tagArray = array('comedy', 'thriller', 'horror');
$videos = Video::whereHas('tags', function($tagQuery){
$tagQuery->whereIn('title', $tagArray);
})->get();

Firtzberg said:

If every video must have all the tags at the same time, then use this

$tagArray = array('comedy', 'thriller', 'horror');
$videoQuery = Video::...;
foreach($tagArray  as $tag){
   $videoQuery =$videoQuery ->whereHas('tags', function($tagQuery){
       $tagQuery->where('title', '=', $tag);
   });
}
$videos = $videoQuery->get();

This was exactly what i was looking for! Thank you so much :D