In the days of kings who had absolute power - it was not a wise thing to make the king angry. Even in today's world it is not wise to make the chief executive in any situation angry enough to have "wrath."
The proverb warns us that the wrath of the king is like the roaring of a lion. A roaring lion is rather intimidating - and tends to make most other animals in the wild freeze with fear. Someone who makes the king rise to wrath is someone who probably did not have enough respect for him in the first place. When we think biblically about a king's wrath - we should go to the book of Esther. There we watch a man fall from the king's closest association - to having a death sentence pronounced upon him. Haman's sin was that of believing his own press a little too much. He was raised up to be the king's most trusted advisor - but used his position and power to carry out a vendetta against a man who did not stroke his ego like he wanted. Haman bristled when Modecai, Esther's adoptive father, would not bow in reverence to him. Without thinking about anything other than his own wounded pride, he vowed within himself to destroy both Haman and his people, the Jewish nation. He did not think enough, because had he done his homework, he would have learned that Esther the queen was in that group he wanted slaughtered. Focused on himself - and blind to anything other than his own pride - he forged ahead with his plan to bring death on the Jews.
In the end, Haman's plan brought death to himself and all others who had a vendetta against God's people. His last act of begging for mercy at the couch of queen Esther so enraged the king that Haman's head was covered in a black bag, in preparation for his hanging. His risked the wrath of two kings - both Ahasuerus on earth - and God the King of heaven and earth. Ahasuerus rendered his judgment on earth - but only did so because God worked sovereignly to bring information to bear on the situation in perfect timing.
The other side of this proverb is that the favor of the king is like the dew. The dew is a wonderful daily manifestation of nutrient-filled water that comes as a result of condensation that happens every evening as cold meets hot in the atmosphere. It causes a plant to thrive and grow. So having the king's favor is a means of us growing and thriving as well. This is true on earth - but has its greatest application with the God of heaven. That favor comes through the gospel of Jesus Christ - which brings God's favor upon us. When we know God's favor through the gospel - we have his daily blessing of dew - as His glory meets our need - as His power fills us and gives us all that we need to overcome sin and grow.
Wisdom knows not to agravate the king - but to know his favor through loving and obeying Him. That same wisdom will also guide us to seek God's favor through the gospel of His grace - so His favor will rest upon us. What an awesome thing though to grasp fully that His favor is given due to Christ's accomplishments - and never due to our own.