I've recently been introduced to Kali Linux's webkiller tool, located in the websploit category. It works on the concept of Tcpkill.
I don't understand how the DoS attack with Tcpkill works. Can someone explain it in detail?
I think you may be misunderstanding what tcpkill is. tcpkill is not generally used in any sort of DDoS type of attack. It is possible to use it to DoS someone, but that usually involves you being within the same network segment as them, or controlling a part of the network their data transits.
To keep it short, its a packet filter. It is designed to drop packets based on your filter criteria, which can be any of the following:
The idea of webkiller is to deny web service to a user on a controlled network. That may be legitimate control, like a network admin blocking certain services. Or it might be illegitimate control, like an attacker blocking access to facebook to piss someone off.