How to call a WP Class inside my theme

Question

You should be able to do something like this:

$wp_customize = new \WP_Customize_Manager();

…since, as I understand it, WordPress core puts all its classes, functions, etc. in the global namespace.

Edit

It appears that $wp_customize may be a bad example, as it’s a global variable (so, in your theme, all you should need to do is declare global $wp_customize; before you use it).

If you’re looking to learn about how to use the Customize API, I’d recommend you read through the official Customize API documentation. Note that, anywhere you need to call a WordPress class or function, you can do so by prepending a \ to its name.