I use ~ in the URL and thumbnails don’t load

This happens because URLs with a tilde symbol (~) are considered temporary while you haven't connected to a domain yet. On some servers this is problematic because the directory on the server itself doesn't contain that symbol, thus TimThumb cannot find your image internally. It would treat it as an external image, but it stops at an error because in some cases the document root is misconfigured as well. Note that this error will go away once you get a domain or your staging area is on a subfolder of that domain and does not look like this: http://192.168.1.100/~something/ which is again, temporary. In the meantime, here is how to solve the issue:

1. Create this file anywhere on your WordPress installation:
   
   <?php phpinfo(); ?>

   1 2 3

   <?php phpinfo(); ?>

2. Name it anything you like, such as info.php and open it with your browser.
3. Look into the Environment section > SCRIPT_FILENAME  and note the value, it's something like/home/username/public_html/wp-content/plugins/justified-image-grid/pi.php - copy it.
4. Open /wp-content/plugins/justified-image-grid/timthumb-config.php file via FTP and add this line after the opening<?php  tag:
   
   $_SERVER['DOCUMENT_ROOT'] = '/home/username/public_html/';

   1	$_SERVER['DOCUMENT_ROOT'] = '/home/username/public_html/';

   Your value will be different, and it's what you copied from the information table. However remove everything that starts with wp-content. End it with a slash. It's often no problem if the whole thing is just a single slash (/).
5. Delete the info.php file you created, it's no longer needed.