upload.php

<?php
function upload_user_file( $file = array() ) {

	require_once( ABSPATH . 'wp-admin/includes/admin.php' );

      $file_return = wp_handle_upload( $file, array('test_form' => false ) );

      if( isset( $file_return['error'] ) || isset( $file_return['upload_error_handler'] ) ) {
          return false;
      } else {

          $filename = $file_return['file'];

          $attachment = array(
              'post_mime_type' => $file_return['type'],
              'post_title' => preg_replace( '/\.[^.]+$/', '', basename( $filename ) ),
              'post_content' => '',
              'post_status' => 'inherit',
              'guid' => $file_return['url']
          );

          $attachment_id = wp_insert_attachment( $attachment, $file_return['url'] );

          require_once(ABSPATH . 'wp-admin/includes/image.php');
          $attachment_data = wp_generate_attachment_metadata( $attachment_id, $filename );
          wp_update_attachment_metadata( $attachment_id, $attachment_data );

          if( 0 < intval( $attachment_id ) ) {
          	return $attachment_id;
          }
      }

      return false;
}
?>

usage.php

<?php
if( ! empty( $_FILES ) ) {
  foreach( $_FILES as $file ) {
    if( is_array( $file ) ) {
      $attachment_id = upload_user_file( $file );
    }
  }
}
?>