What does Proverbs 27:2 mean?

The prior proverb noted the arrogance of bragging about things a person has not yet accomplished (Proverbs 27:1). This teaching criticizes those who brag about things they have accomplished. In both cases, the concern is arrogance. Or, an inflated sense of one's own greatness. Even if the statements are factual, crowing about oneself doesn't look good to others. The better path is to let others speak well of you; it seems more like sincere praise and less like bragging.

In Daniel 4 we learn that King Nebuchadnezzar foolishly bragged about his past victories. He was walking on the palace roof and viewing the city of Babylon when he sang his own praises. He boasted: "Is not this great Babylon, which I have built by my mighty power as a royal residence and for the glory of my majesty?" (Daniel 4:30). Before he was even done speaking, a heavenly voice declared his impending humiliation. That would include insanity, wallowing with animals, and eating grass. For seven years Nebuchadnezzar endured this, until he fully understood "that the Most High rules the kingdom of men and gives it to whom he will" (Daniel 4:31–34).