The url you are trying isn't working because the index file is inside the "public" directory. So if you just manualy remove it from the URL it's expected to fail.

To remove the "public" part from the url you will have to create a Virtual host, which will point to the public directory.

If you are doing this locally, it's easy as:

1. open terminal

2. Edit host file

sudo gedit /etc/hosts

3. add line:

127.0.0.1   app.dev

then save and close.

4. Edit server config

sudo gedit /etc/apache2/sites-available/000-default.conf

And add something like

<VirtualHost *:80>

	ServerName app.dev
	DocumentRoot /path/to/laravel/public

</VirtualHost>

5. Make sure to restart apache

sudo service apache2 restart

Thanks for your trouble but unfortunately it is not that simple. Synology implemented quite a few changes to the Linux install and not all of them are clear or well documented (if you want to do something out of the ordinary). I have searched long and hard and still haven't found a complete and bullet proof explanation on how to get a VirtualHost up and running on a Synology NAS (at least not for DSM 5).