$instance: how does it know the info of badges?

I think you may be mixing up what data is stored in $instance and what data is stored via the Options API, and I think my first answer may not have helped at all in that case.

$instance are options that pertain to the specific widget that is in use. Say you have 3 or 4 of these widget throughout your site, well that $instance is not universal to all of them. When you update the one of the widget forms, the function will run for that widget and the data will run through $instance as you see it in your functions, but it only pertains to that specific instance of the widget. This is much more the Widgets API than the Options API at work here. This is why the only thing you would have stored via $instance would be options that you need for the presentation of the widget as it will display on the frontend, all for the sake of the individual widget, and not for all widgets of the same type.

What you store via the options page and via the Options API functions is data that is universal to all Badges widgets in use on the site. So, $instance has nothing to do with it, and will never actually interact with it. What happens is that the options stored in $instance are used in the code to conditionally handle how we present the a specific widget.

It's also important to remember that the option of wptreehouse_badges is just the name of where the options for this plugin are accessible via the Options API. This doesn't mean that badges are stored there. From what I can see in the plugin's wptreehouse_badges_options_page() function, $options['wptreehouse_profile'] is actually storing the entire JSON feed retrieved from the AJAX call. So, by accessing that variable alone, you have the ability to traverse the object and get whatever data that exists in the stream.

This is actually happening in the widget method:

<?php
function widget( $args, $instance ) {
    // Widget output.

    extract( $args );
    $title = apply_filters( 'widget_title', $instance['title'] );
    $num_badges = $instance['num_badges'];
    $display_tooltips = $instance['display_tooltips'];

    // Right here you can see we are getting all of the data stored in `wptreehouse_badges`.
    $options = get_option( 'wptreehouse_badges' );

    // And here we are assigning the entire JSON stream in the retrieved options data in `$wptreehouse_profile`.
    $wptreehouse_profile = $options['wptreehouse_profile'];

    require( 'inc/front-end.php' );

}

Hopefully this helps a bit. Let me know if you have any more questions, or just if my explanation is no help at all, lol.