How do the man behind the desk and Eckels describe the opponent of the newly-elected president? Why is this part of the narrative significant in "A...

The man behind the desk and Eckels describe the opponent of their new president as a man who is against everything. They are both relieved that the president is another man. This part of the narrative is significant because it acts as foreshadowing of what is to come in the short story.

After Eckels enters the office of Time Safari, Inc., he chats briefly with the man behind the desk about the ramifications of time travel: 

"Makes you think, If the election had gone badly yesterday, I might be here now running away from the results. Thank God Keith won. He'll make a fine  President of the United States."

The clerk concurs,

If Deutscher had gotten in, we'd have the  worst kind of dictatorship. There's an anti everything man for you, a militarist, anti­-Christ, anti­-human, anti­-intellectual.

Ironically, however, after Eckels, who has paid thousands of dollars to travel back in time sixty million years in order to shoot a Tyrannosaurus Rex, becomes too frightened to shoot this mammoth and forbidding animal; instead, he flees and slips off the anti-gravity path. He commits this action despite having been told by the guide Travis to not touch anything in this world of the past because

[A] little error here would multiply  in sixty million years, all out of proportion.

Indeed, radical change is what occurs because when Eckels reboards the ship, he has mud on the bottom of his shoes. One of the guides worried about fines and other negative repercussions, makes Eckels find the bullets in the dead monster so that there is no proof of their having been there. But, neither the other men nor Eckels realize that a small, dead butterfly lies in the mud of the sole of his boot. In fact, it is not until Eckels returns and finds a sign which he cannot read that he discovers the butterfly on the bottom of the boot. Incredulous that this insect has caused alterations in present time, Eckels asks the new clerk who won the presidency. The man replies,

"You joking? You know very well. Deutscher, of course! Who  else? Not that fool weakling Keith. We got an iron man now, a man with guts."

What the original man behind the desk has mentioned as a horrible possibility now has come true.