I was wondering what kind of technologies are behind Web Push, for example https://goroost.com
How is that working? How to configure my server for Web Push? What should I look at?
The underlying technology is the Push API, combined with the Notifications API
You can read more about how to implement this yourself in Google's developer docs. Note that currently push notifications are only supported in Google Chrome, but Microsoft and Mozilla have said they will be implementing it soon.