How to remove woocommerce added cart items and redirect to checkout?

Solution:

First you will need in WooCommerce Settings > Products > Display in “Add to cart behaviour” to enabled the checkbox : Redirect to the cart page after successful addition

Then you will need the 3 following hooked function:

1) Empty cart before add-to-cart (if cart is not empty)

add_filter( 'woocommerce_add_to_cart_validation', 'one_cart_item_at_the_time', 10, 3 );
function one_cart_item_at_the_time( $passed, $product_id, $quantity ) {
    if( ! WC()->cart->is_empty())
        WC()->cart->empty_cart();
    return $passed;
}

2) Add-to-cart redirection to checkout:

add_filter( 'woocommerce_add_to_cart_redirect', 'add_to_cart_checkout_redirection', 10, 1 );
function add_to_cart_checkout_redirection( $url ) {
    return wc_get_checkout_url();
}

3) Skip cart page redirecting to checkout:

add_action('template_redirect', 'skip_cart_page_redirection_to_checkout');
function skip_cart_page_redirection_to_checkout() {
    if( is_cart() )
        wp_redirect( wc_get_checkout_url() );
}

Code goes in function.php file of your active child theme (or active theme).

Tested and works (with Woocommerce 3.2.6 and WordPress 4.9.2 on Storefront theme).