Anything that comes out of this query– $event = Ai1ec_Events_Helper::get_event($post->ID);— will need the $event-> part. That is how you access data from an object. That is just pure PHP.
I don’t know why some of the values in that answer, like $contact, are not prefixed with $event-> but I am assuming those variables do not come from from that query.
Add var_dump($event); immediately after $event = Ai1ec_Events_Helper::get_event($post->ID); and you can see what information is in that object.