I heard this on a TV show and I've never really heard a sentence similar to it before. I'd appreciate it if you would help me out with the meaning of the sentence and its structure.
"Having the last word" means making the final statement in an argument between two people. Obviously every argument has to end, and so someone has to have the last word, but it can be difficult to let the other person have the last word, as it feels like the person who has the last word has "won".
If someone "has to have the last word", it means that they can't finish an argument by listening to the other person. A person who has to have the last word always has to "win", they can't compromise
"Its just like him" is a way of talking about a person's character by describing their typical behaviour:
He's really rude. It's just like him to leave the dinner table before others have finished eating.
He's so kind. It's just like him to help the old lady who lives next door.
So this is saying talking about his character by saying he usually has to have the last word in an argument. In other words, he is the sort of person that usually wins arguments. He must have a very forceful personality. But is unable to compromise or listen to other people's viewpoint. It means they are obstinate and can't back down, even when they are clearly wrong.
This is a negative thing to say about a person.