As you rise in power, attention will inevitably turn on you. The book says that, in these cases, you should not take all the credit, but act in such a way that your "audience" will see that your achievement is "attainable" as well. It is just not natural for people to be too perfect, and appearing so will create envy in other people. This envy will make them turn either hypocritical, or over-praising; both indications of upcoming trouble. You should acknowledge some shortcomings to appear more human, to make people see you as this real and approachable person. In short, never take all the credit for your accomplishments. Say that luck or other factors also had a play in your success.
An example of a person who may relate to this law is Joseph. When he told his dream to his brothers, it was not exactly an "achievement" yet, but as he is already envied by his brothers, his dreams of success led to more envy, which became the cause of his demise. He redeemed himself, though, in the future, when he was already a vizier. He said that it was in the LORD's plan to save their family that they [his brothers] sold him to the passing traders (sort of emphasizing the "fate" part that he is in that position).
But in this age, where people are generally more competitive, and want to project that he/she is always better than the next person, what are the pitfalls of admitting to your weaknesses?