This proverb has a lot to do with past ideologies. Gone were the days when carrying a stick was one of the characteristics of old age. And, normally, the aged were regarded as wise.
Speak softly and carry a big: People who speak softly are regarded as wise, nice, peaceful, and peacemakers. Speaking softly, coupled with holding a big still, makes you earn a great deal of respect and favor in the sight of the people for they highly regard you.
You will go far: You will go far because you will be respected by both young and old, you would be considered for positions, your council would often be sought concerning important matters and, generally, trouble would be far from your doorstep – for why would anyone want to trouble a wise peace lover?
I think even today, such a strategy works, except for the carrying of big sticks. Normally, you reap what you sow. It is a virtuous thing to speak softly to people. If you treat people kindly or softly, they will treat you the same way in return. And if you treat people harshly, they will treat you the same way too. Rude people piss off everyone.
It all comes down to our social life; the way we relate to others and react to issues. Boastful, rude, and loud utterances when dealing with people only repels friendship, favor, and acceptance. Let try as much as possible to treat one another tenderly, with kind words and gentle body language.
