Custom Meta Box returns no HTML

Try adding array( $this, 'myskills_html' ) to the third argument.

    public function skillsmetabox_init() {
        add_meta_box( 'myskills_meta', 'Proficiency', array( $this, 'myskills_html' ) );
    }

Since this is class method, it should have the $this as well.

The same way you did with

add_action( 'save_post', array( $this, 'save_metabox_details' ) );