what does "walk softly"mean?
【A great man must walk softly. Presidents must yield little points to win big ones, and really, this point was very small, a half-forgotten fico. The specimen was hung, not in the oval office, but in the nearby Fish Room. The President had granted a favor—how great a favor only the First Lady knew—and his Vice President had been genuinely pleased. Give and take; win a few, lose a few; that was the strategy of public office. In one version of his pet joke, the last line of which could be adapted to the occasion, he used to tell his friends that "The three most overrated things in the world are the state of Texas, the FBI, and hunting trophies." Nevertheless the memory of the incident was distasteful.】

1.what does "walk softly"mean?
2.what does "pet joke"mean?

Question

what does "walk softly"mean?
【A great man must walk softly. Presidents must yield little points to win big ones, and really, this point was very small, a half-forgotten fico. The specimen was hung, not in the oval office, but in the nearby Fish Room. The President had granted a favor—how great a favor only the First Lady knew—and his Vice President had been genuinely pleased. Give and take; win a few, lose a few; that was the strategy of public office. In one version of his pet joke, the last line of which could be adapted to the occasion, he used to tell his friends that "The three most overrated things in the world are the state of Texas, the FBI, and hunting trophies." Nevertheless the memory of the incident was distasteful.】

1.what does "walk softly"mean?
2.what does "pet joke"mean?

Answer

Walk softly means that the president chose to avoid an argument with Johnson over the small matter of the hunting trip and and the hanging of the trophy, even though he found both appalling.  A good leader isn't afraid to lose small battles if it helps him win bigger battles at a later time.  
Pet joke means it was an inside joke among like-minded friends.  This was a way of mocking Lyndon Johnson and would have never been told while Johnson was around them.

Answer

To walk softly means to not disagree with people very often.