WP_list_table prepare_items()

You are processing the New Terms after you’ve displayed the Table List already, so in order to display the new term, as soon as you submit the form, you’d need to process it before you display the table.

I’ve added a new function process_form() in here, that handles the processing of new_trem form submission, so when the form is submitted, it’ll verify the nonce field ( Which you should always use wp_create_nonce_field() ) and add a term.

Also, for page headings you don’t need the icon32 div anymore <div id="icon-users" class="icon32"><br/></div>, you should remove this one.

And the page headings need to be enclosed in <h1> tags for admin screens since WordPress 4.3: https://make.wordpress.org/core/2015/07/31/headings-in-admin-screens-change-in-wordpress-4-3/

function tt_render_list_page() {
    //Handle New Term Form Submission
    $this->wpse_220820_process_form();
    //Create an instance of our package class...
    $testListTable = new TT_Example_List_Table();
    //Fetch, prepare, sort, and filter our data...
    $testListTable->prepare_items(); ?>
    <div class="wrap">

        <div id="icon-users" class="icon32"><br/></div>
        <h1>List Table Test</h1>

        <form id="movies-filter" method="GET">

            <!-- For plugins, we also need to ensure that the form posts back to our current page -->
            <input type="hidden" name="page" value="<?php echo $_REQUEST['page'] ?>"/>
            <!-- Now we can render the completed list table -->
            <?php $testListTable->display(); ?>

        </form>
        <form id="insert_term" name="insert_term" method="post" action="">

            <div class="form-field  term-name-wrap">
                <label for="term">Term</label>
                <input type="text" value="" name="term" id="term" size="40"/>
                <p>Description</p>
            </div>

            <label>Description</label><input type="text" value="" name="termdesc" id="termdesc"/>

            <input type="submit" value="Add term" id="submit" name="submit"/>
            <input type="hidden" name="action" value="new_term"/>
            <?php wp_nonce_field( 'wpse_220788_new_term', 'new_term_nonce' ); ?>
        </form>
    </div>
    <?php
}

/**
 * Checks $_POST for Form submission and insert it into table
 */
function wpse_220820_process_form() {
    if ( ! empty( $_POST['action'] ) && $_POST['action'] == "new_term" ) {
        //Verify Nonce
        if ( ! empty( $_POST['new_term_nonce'] ) && wp_verify_nonce( $_POST['new_term_nonce'], 'wpse_220788_new_term' ) ) {
            if ( isset( $_POST['term'] ) && ! empty( $_POST['term'] ) ) {

                $new_term    = sanitize_text_field( $_POST['term'] );
                $description = sanitize_text_field( $_POST['desc'] );
                wp_insert_term(
                    $new_term,
                    'productcat',
                    array(
                        'description' => $description,
                    )
                );

            }
        }

    }
}