Why does not my metabox save?

I notice a couple of things.

First, your callback uses two parameters but you don’t ask for the second parameter when you hook it in. This:

add_action( 'save_post', 'kasparibi_left_menu_meta_save' );

Should be:

add_action( 'save_post', 'kasparibi_left_menu_meta_save' ,1 ,2 );

But, on the other hand, I don’t see where you use that parameter at all. Maybe I missed it. You could probably just leave that second parameter out of the callback.

The real problem is here though:

<?php (get_post_meta( $post->ID, 'left_menu_checkbox', true) == 'on') ? ' checked="checked"' : ''; ?>

You don’t echo the checked attribute.

<?php echo (get_post_meta( $post->ID, 'left_menu_checkbox', true) == 'on') ? ' checked="checked"' : ''; ?>

In other words, the value does save, but your code does not display the checked status correctly.

You could simplify that line with WordPress’s checked function though.

<?php checked(get_post_meta( $post->ID, 'left_menu_checkbox', true),'on',true) ?>