To exclude a certain Category’s posts from the home page you just need to know the category ID and then use a filter on the pre_get_posts hook to exclude those posts.

To find the ID for the Category just go to the Category and hover over the edit button and the ID appears in the URL at the bottom.

//Exclude Category Posts from Home Page
function themeprefix_exclude_category($query) {
	if ( $query->is_home() ) {
	$query->set('cat', '-338');//add your category number
	}
	return $query;
}
add_filter('pre_get_posts', 'themeprefix_exclude_category');

Then add the function inside your functions.php file. The above function only runs on the home page and will remove any posts that belong in the category numbers listed. In the exmple above that would be number 338, to exclude more than one category just comma separate all the ID numbers like so; ‘-338, -340, -350’