I think that's a really good question. Why do we try to convert others to our way of thinking?
Maybe some people are threatened by other viewpoints, indeed. But that's probably not the main reason. (Anyway, mostly people whose way of thinking threatens others - terrorists, say, or Vladimir Putin - will likely be immune to converting arguments. Sadly.)
So why, then? Well, for millennia, people have found it useful to argue the toss about important matters as a way of coming to see the truth.
... Suppose I think so-and-so, whereas you think such-and-such. I might be wrong. So, to check it out, I argue with you (I argue with myself too, in a way, but that seems to be less fruitful): "Here's why so-and-so", I say; "Ah, but have you considered such-and-such", you reply; "Hmm, interesting," I go, "But, then, if such-and-such, doesn't it follow that this-and-that"; ... and so on and so on, hopefully not ad infinitum.
Perhaps one of us might be convinced. Perhaps not. But the very process of arguing like this - trying to convert another person, while listening and responding to their attempts to convert you, is likely to expose flaws, logical and otherwise, in the disputants' thoughts, and so to conduce, in the end, to knowledge of what is actually the case.
This process was named 'dialectic' by the old Greek philosopher Plato. The term has had a long and distinguished career since then, being used in different contexts and with different connotations. However, I can report, it works! Given a certain amount of good will and respect on all sides, attempted conversion does often help us to see the truth - or, perhaps less ambitiously, at least to unmask falsehood.
Try it!