Daemon killed Vaemond because he called Rhaenyra's children bastards right in front of their mother and King Viserys himself. In episode 7, Viserys said that he would take the tongue of anyone who called the children bastards. As such, Daemon took it upon himself to kill Vaemond right then and there.
It is also interesting to note that Daemon and Vaemond never actually liked one another from the very start. As such, that was one of the reasons that made Daemon want to kill Vaemond. It was only the fact that Vaemond called Rhaenyra's children bastards that Daemon had every reason to take his head. That said, let's talk more about why Daemon killed Vaemond.
Daemon Killed Vaemond | House of the dragon EP-8
Daemon Killed Vaemond | House of the dragon EP-8
Daemon Killed Vaemond | House of the dragon EP-8
Daemon Killed Vaemond: House of the dragon EP-8
Daemon Killed Vaemond: House of the dragon EP-8
Daemon Killed Vaemond: House of the dragon EP-8
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