The application is written on Laravel 5.0. Now I would like to "install" my web application on several environments (means several servers). But always copying the changed files seems too costly for me. I'd like to only have to trigger maybe a script, and all applications on all servers will be updated (except the config files).
Maybe something like that is possible with git?
Yes. You can surely do it using git but it will only fetch your code only not server configuration. For the laravel installation and your coding: Just make a git repository and upload your code over there and just pull the code from git remote command.
But, It will not auto-configure your server. Like installing composer, mcrypt, lamp stack and so on. That you will need to configure.