Custom page templates are for pages – not post type archives 🙂
Naming your template file as archive-projects.php will cause WordPress to select it for your Projects archive page.
You also don’t need the /* Template Name: Projects */ in there, as this template naming is default WordPress behaviour. See the visual overview of the template hierarchy to understand how you can name your template files so WordPress will select them.
Alternatively, if you are wanting to replace the default WordPress custom post type archive with your own page, just ensure your page has the same slug as the custom post type – in this case, that would be projects. You can change the slug of the page at the top of the post editor, just under the title.