It's a rather archaic thing to be speaking of listening these days, I know. I mean, we are all engaged in shouting as loud as we can, expecting to go viral, that we hardly have time to listen to anyone. I mean, it's hard enough to hear yourself think above the din of your own shouting so where's the scope to listen to anyone else? So, yes, this is one of those pieces where we can all go, "So that's how people lived in the old days! How quaint!"
So, what Tiru says in this Kural probably made more sense in his times.
Nunangiya kelviyarallaar vanangiya vaayinar aadhal aridhu - Tirukkural
He who is not a discerning listener very seldom manages to be a polite speaker - Loose Translation
You know, it's very tempting to just understand it as 'You cannot be a capable speaker if you do not have the habit of listening', which is ONE way to interpret this couplet. I mean, after all, if you are being a convincing speaker and are being listened to with respect, you have no need to yell and shout and call people names, all of which will be counter-productive, so you WILL be polite. AND to be a such a speaker you need to have the habit of listening with discernment to all that you hear so that you can not only learn well from what people say but can also understand what they are saying. Including the discernment of what is right in it and what is wrong.
You can see it the other way. IF you are incapable of discernment while listening, you truly do not understand what is being said. In that case, and especially if you are in a debate or argument, you cannot speak politely because the hollowness of your argument is more clearly visible in calm discussions. You tend to bluster and yell so that you can win your point by the sheer volume of your voice. So, yes, if you are not a discerning listener, it is tough to be a polite speaker.
Of course, there is this problem as well of not truly understanding what the other person has said, because you did not properly listen, and getting angry because you misunderstand his position totally and go into a screaming fit. More often than not, discussions turn into screaming matches because one person fails to listen to the other properly and starts attacking him personally (what we call ad hominem attacks) because he cannot attack the other guy's points since he did not even understand them. (Oh, yeah, I AM talking about well-meaning discussions. Ad hominem attacks CAN also happen because the listener understood the point all too well, has no counter to it, and tries to win the discussion by converting the debate into a quarrel!)
So, you see, Tiru can convey too many things with just a couple of lines. Though, yes, about archaic abilities like listening which, apparently, Homo Sapiens once had!