@StephenOstermiller's answer has done a great job of describing the .htaccess file / front-controller process, but I thought I'd address the other issue you raised...
If I comment out the things in the ".htaccess" file, nothing changes
That's because the format of the URLs you are using (eg. /ROUTER/index.php/home) completely negates the need for the .htaccess file to begin with. You are calling index.php (the "front-controller") directly in the URL itself and passing /home as additional pathname information (aka. path-info).
The .htaccess file is still processed, but the 2nd condition (RewriteCond directive), which checks that the request does not map to a physical file, fails (index.php is a physical file) so the rule is not triggered (it does nothing).
The additional path-info on the URL is available to PHP in the $_SERVER['PATH_INFO'] superglobal. So, if you are using URLs of the form /ROUTER/index.php/home then you could write your front-controller (index.php) like this instead (simplified):
// URLs of the form "/ROUTER/index.php/home"
$request = $_SERVER['PATH_INFO'];
switch ($request) {
    case '':
    case '/home':
         :
    case '/about' :
         :
}
(As noted above, this is not making use of .htaccess)
On the other hand, your .htaccess file allows you to have URLs of the form /ROUTER/home (or simply /home), avoiding you having to include index.php in the URL, which is then internally rewritten to index.php (by the rule in .htaccess). You then use $_SERVER['REQUEST_URI'] (as in your original script) to access the requested URL in PHP.
For example (simplified):
// URLs of the form "/ROUTER/home"
$request = $_SERVER['REQUEST_URI'];
switch ($request) {
    case '/ROUTER/':
    case '/ROUTER/home':
         :
    case '/ROUTER/about' :
         :
}
However, your existing .htaccess file is not configured correctly for this. The .htaccess file is assuming index.php is located in the document root, but your URLs suggest you have a /ROUTER subdirectory, in which index.php (the "front-controller") is located.
If your .htaccess file is in the /ROUTER subdirectory at /ROUTER/.htaccess then remove the RewriteBase directive entirely.
If, however, your .htaccess file is located in the document root then you will need to change your RewriteBase directive to read:
RewriteBase /ROUTER
(Setting RewriteBase /ROUTER in both cases will also work.)