By default bbpress can be quite strict on the HTML it allows in posts (tags are whitelisted). The following loosens those restrictions and is taken from this post: http://buddypress.org/support/topic/tutorial-allow-more-html-tags-in-bp-forum-topics/

bbpress-kses.php

<?php
function myprefix_kses_allowed_tags($input){
    return array_merge( $input, array(
          // paragraphs
          'p' => array(
               'style'     => array()
          ),
          'span' => array(
               'style'     => array()
          ),

          // Links
          'a' => array(
               'href'     => array(),
               'title'    => array(),
               'rel'      => array()
          ),

          // Quotes
               'blockquote'   => array(
               'cite'     => array()
          ),

          // Code
          'code'         => array(),
          'pre'          => array(),

          // Formatting
          'em'           => array(),
          'strong'       => array(),
          'del'          => array( 'datetime' => true, ),

          // Lists
          'ul'           => array(),
          'ol'           => array( 'start'    => true, ),
          'li'           => array(),

          // Images
          'img'          => array(
               'src'      => true,
               'border'   => true,
               'alt'      => true,
               'height'   => true,
               'width'    => true,
          )
     ));
}
add_filter( 'bbp_kses_allowed_tags', 'myprefix_kses_allowed_tags', 999, 1 );
?>

How can I add support for videos?

See:

https://bbpress.org/forums/topic/extra-tags-to-add-to-list-for-videos/#post-205701

I added:

'video'      => array(
	'controls' => true,
	'width'    => true,
	'height'   => true,
),
'source'     => array(
	'src'    => true
)

But it is not right. The source inner component gets stripped.