I think a lot of the downplaying of talents/achievements comes from the fact that more intelligent people tend to downplay their own achievements, whereas less intelligent people tend to overplay their own achievements. This definitely applies to me, as whilst I've never perceived myself to be very intelligent/skilled and have tended to focus, to my own detriment, on what I am not skilled at (not helped by suffering from depression for around a year or two but I won't get into that).
Personally I have tended to be humble and downplay my achievements, as I never really have wanted to sound arrogant ever, but I have found that its good to take pride at what you are good at, but also to aspire to become better at what you are good at (and accept you can't be good at everything - which took me a while to come to terms with). Also found that, even when downplaying my skills in front of good friends, it usually, at least externally (i can't speak for what they actually think) brings out positivity and that has increased my own self-esteem, as naturally most crave to be validated by our circle of friends/family and use that to plant confidence in ourselves.
I agree that society tends to disapprove on those who hold themselves in "too" high self-esteem, but sometimes with good reason. Nonetheless having very little self-esteem is not good and no-one should aspire to have a low self-esteem, as I have suffered from it and it can become a crippling weakness if not dealt with.
This post is kind of a mess of jumbled thoughts in a short post but I don't have time to reorder it as by writing this I'm procrastinating on other stuff I should be doing