Rudimentary example of taking form data and using it to create a (bbPress) forum topic

simple-form-to-forum-post.php

<?php
function post_to_bbp_shortcode() {
	if ( 'success' === post_to_bbp_status() ) {
		return post_to_bbp_success();
	}
	else {
		return post_to_bbp_form();
	}
}

function post_to_bbp_form() {
	$check = wp_create_nonce( 'post_to_bbp' );

	return "
		<form method='post'>
			<input name='post_to_bbp_msg' />
			<input type='submit' value='Send!' />
			<input type='hidden' name='post_to_bbp_check' value='$check' />
		</form>
	";
}

function post_to_bbp_success() {
	return 'Thank you for your submission.';
}

function post_to_bbp_listener() {
	if ( ! isset( $_POST['post_to_bbp_check'] ) || ! isset( $_POST['post_to_bbp_msg'] ) ) {
		return;
	}

	if ( ! wp_verify_nonce( $_POST['post_to_bbp_check'], 'post_to_bbp' ) ) {
		return;
	}

	bbp_insert_topic( array(
		'post_parent'  => 4, // This needs to be a valid forum ID!
		'post_title'   => 'Result of submission dated ' . date_i18n( 'Y-m-d H:i:s' ),
		'post_content' => filter_var( $_POST['post_to_bbp_msg'], FILTER_SANITIZE_STRING )
	) );

	post_to_bbp_status( 'success' );
}

function post_to_bbp_status( $value = null ) {
	static $status;

	if ( null === $value ) {
		return $status;
	}
	else {
		$status = $value;
	}
}

add_action( 'init', 'post_to_bbp_listener' );
add_shortcode( 'post_to_bbp', 'post_to_bbp_shortcode' );

This is a purely illustrative example! Notes:

• The shortcode [post_to_bbp] should be used wherever the form needs to be generated
• It doesn't take care of sanitizing the input or guard against repeat submissions

It really just shows a basic approach to creating a custom form that generates forum topics; in general the [bbp-topic-form] shortcode that ships with bbPress may well be a better bet/starting point for your own customizations.