Display categories using get_the_terms();

archive.php

// Get terms for post
$terms = get_the_terms( $event->ID , 'tribe_events_cat' );

// init counter
$i = 1;

// Loop over each item since it's an array
if ( $terms != null ){
  foreach( $terms as $term ) {
  // Print the name method from $term which is an OBJECT
  print $term->name;
  //  Add comma (except after the last theme)
  echo ($i < count($terms))? ", " : "";
  // Increment counter
  $i++;
  // Get rid of the other data stored in the object, since its not needed
  unset($term);
  }
}