How are birds related to T-Rex if the key Rex was one of the last dinosaur to be killed?

They are not very closely related to Tyrannosaurus. All carnivorous dinosaurs, including Tyrannosaurus belonged to a group called theropods. Birds evolved from a group of small theropods related to the dinosaurs colloquially called raptors. Tyrannosaurs were a distinct group of theropods from the one that produced birds.