What does "Rewrite URL structure" do?

What does Minify>Rewrite URL structure do, actually?

I am not the developper, but I will try to respond to the best of my knowledge. When “Rewrite URL structure” is disabled, the URL of minified CSS and JS files looks like this:

wp-content/w3tc/min/index.php?file=whatever.css
wp-content/w3tc/min/index.php?file=whatever.js

When “Rewrite URL structure” is enabled, the following line is added in wp-content/w3tc/min/.htaccess :

RewriteRule ^(.+\.(css|js))$ index.php?file=$1 [L]

This rewrite rule allows to hide the “index.php?file=” part, and to have a nice-looking URL for minified CSS and JS files :

wp-content/w3tc/min/whatever.css
wp-content/w3tc/min/whatever.js

That’s all.

And, how does it relate to the end slash issue (.com/) ?

I’ve never experienced such issue, so I can’t help. It sounds unrelated.