Removing the shipping cost from shopping cart can be easily achieved in WooCommerce by using filters hook.

// Step 1: Remove shipping cost from CART

add_filter( 'woocommerce_cart_ready_to_calc_shipping', 'devnodes_disable_shipping_on_cart' );
add_filter( 'woocommerce_cart_needs_shipping', 'devnodes_disable_shipping_on_cart' );
function devnodes_disable_shipping_on_cart( $enabled ){
    return is_checkout() ? true : false;
}

// Step 2: optional, Let the user know that shipping calculated at checkout
add_action('woocommerce_after_cart_table', 'devnodes_after_cart_contents');
function devnodes_after_cart_contents() {
    echo '<div class="woocommerce-info" style="border-color: #ddd;" role="alert">Shipping charge calculated at checkout</div>';
}