The button on the browser just navigates to the URL `git-peek://https://github.com/name/repo`. How your system handles this git-peek protocol is completely up to you. While the git-peek package does offer to setup a handler for this custom git-peek protocol, I went ahead and set it up manually. Now, my system calls this bash script whenever it encounters the git-peek protocol:
I tend to clone things I'm just having a quick look around in into ~/tmp - sometimes I'm intending to spelunk through the history so I don't fancy having that much data sat on a tmpfs and "running rm -rf ~/tmp/* when I notice it's getting a bit on the large size" is minimal enough effort that it's worth it for having control over when things disappear.
Having it as button in browser seems cool but also horribly insecure...