Basically this plugin switches the theme template that is loaded by WordPress.
“I read somewhere that you can make changes in the customizer. Are those changes live or do you actually have to activate the plugin for them to be live?”
The plugin and the Customizer are two different things. I can answer questions about the plugin. If you have questions about the Customizer (a core feature), you can ask in the general WP support forum.
Also, for more detailed description of how this plugin works, check out the following guides:
The sections titled “What it’s for..” and “What it’s NOT for..” on the homepage:
https://www.ads-software.com/plugins/theme-switcha/
The section titled “How It Works” on the Installation page:
https://www.ads-software.com/plugins/theme-switcha/#installation
The sticky post in the plugin’s support forum:
https://www.ads-software.com/support/topic/important-please-read-2/
And there also are various explanations and infos in the plugin’s FAQs (also on the plugin homepage).
I hope this helps, let me know if I can provide any further information.