Interesting. I had to make a conscious effort to go the other way around, i.e.: to talk more about me. I grew up reading too many books aimed at the extroverts among us ("There's a reason we have 1 mouth and 2 ears!"; "Silence is golden!"). It's hard to underestimate missed opportunities because a shy introvert (me back then) took and (mis)applied advice meant for extroverts who were already unable to stop talking ab…
When growing up I also listened to those statements as you said, in addition I would like to bring out the very common "better to remain silent and be thought a fool than to speak and remove all doubt".
This type of sentiment actually gave me performance anxiety. Thinking people are constantly evaluating you, and at any given time if you speak you could significantly lose that rating. In which case it's always best to stay with the "neutral" or whichever rating. I thought it would be okay to kind of be this "strong, smart, mysteriously silent type".
Unfortunately if you actually follow this strategy, you will be left behind, because people who do talk, and say things that are stupid, are more likely to learn to talk better than anyone staying silent. Eventually they will say fewer stupid things, their communication skills will improve.
If I hold a stupid belief or opinion, and I never voice it, there's no one ever who could make me understand why the belief/opinion is stupid, meanwhile I'm wasting all this time with the bad belief.
These types of advices are, as one would also expect listened only by people who shouldn't listen to the advice, and people who should, won't listen to it in the first place.
So in the end it's people frustrated with other people talking too much giving out this type of advice without any thought on the consequences.
The "better to remain silent..." to me is worst of them all due to the mindset it brings to the social situations. As if it's a constant performance who is able to give a smarter impression, while a better mindset is that social situations and conversations should be about learning, improving, connecting and having fun. Sharing your thoughts, making mistakes, saying stupid things, arguing, are all part of this.
As a teenager I used to perceive most social situations as this sort of competition/performance, so I could never enjoy those situations since naturally the strategy I followed didn't make me become better at the competition. Only way I could've become better is if I actively participated and was okay with losing at least for a while. Now I'm trying to perceive social situations as something where I try to achieve the aforementioned goals (learning, improving, connecting and having fun) for the group as a whole rather than having a competition within the group.
If I ever have a kid and if that kid gets natural disposition to being introvert or on the autistic side, which could be likely due to how genes work, I will make sure to do my best to inject the mindset I believe is more productive.