Source

File: wp-admin/includes/class-custom-image-header.php

public function step_2_manage_upload() {
		$overrides = array( 'test_form' => false );

		$uploaded_file = $_FILES['import'];
		$wp_filetype   = wp_check_filetype_and_ext( $uploaded_file['tmp_name'], $uploaded_file['name'] );
		if ( ! wp_match_mime_types( 'image', $wp_filetype['type'] ) ) {
			wp_die( __( 'The uploaded file is not a valid image. Please try again.' ) );
		}

		$file = wp_handle_upload( $uploaded_file, $overrides );

		if ( isset( $file['error'] ) ) {
			wp_die( $file['error'], __( 'Image Upload Error' ) );
		}

		$url      = $file['url'];
		$type     = $file['type'];
		$file     = $file['file'];
		$filename = wp_basename( $file );

		// Construct the object array.
		$object = array(
			'post_title'     => $filename,
			'post_content'   => $url,
			'post_mime_type' => $type,
			'guid'           => $url,
			'context'        => 'custom-header',
		);

		// Save the data.
		$attachment_id = wp_insert_attachment( $object, $file );
		return compact( 'attachment_id', 'file', 'filename', 'url', 'type' );
	}