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Timothy Garton Ash in The Guardian:
"I have spent the summer in the United States watching, with growing alarm, a country engaged in a degree of self-harming which, if observed in a teenager, would lead any friend to cry 'call the doctor at once'. As I set course back to Europe, my conclusion is this: America should do a reverse Columbus. The world no longer needs to discover America; but America urgently needs to discover the world's view of America."
http://www.theguardian.com/commentisfree/2013/oct/15/americans-need-discover-how-world-sees-them

Well heck, I've got just the prescription: welcome to Their America!

In 2013, a research was conducted by WIN/Gallup to find out which country presents the greatest threat to the global community. About 65,000 people from 65 countries chose the Unites States as the biggest threat to the world peace. America was named the biggest threat by 25% of people; Pakistan, which came second on the poll, was chosen by 8% of people.

Interestingly, the results of the research show that America is perceived as a threat not only by its usual enemies, such as Iraq, Afghanistan and Pakistan, but also by its usual partners from Europe and North America. In Canada, for example, 17% of people see America as the biggest threat to world peace. In Europe, the results are similar. For me, the results are not surprising. I think that America is perceived as dangerous mostly because of its military aggressions around the world, which get a lot of international airtime and show America as greedy and aggressive.

At the same time, the research exposes a paradox: a large number of people who named America as the most dangerous country in the world would pick the United States as their new home if they had to move from their country. I believe that this paradox clearly illustrates the double-sided image of America in the eyes of the world.

Thanks, metak, for jumping in here!
You ask: "why exactly should we care about what Americans think?"
I'm reminded of Pierre Trudeau's famous remark about Canada's relationship to the United States: "Living next to you is in some ways like sleeping with an elephant. No matter how friendly and even-tempered is the beast, if I can call it that, one is affected by every twitch and grunt." Even countries distant from America -- as Yemen, Syria, and now Ukraine (thanks, ivi1303, for embedding that provocative YouTube clip!) have been reminded over the past few months -- share this vulnerability to its movements; it makes sense to know, as it were, which way the beast is rolling, if only to get out of the way. (Zan's point -- about the broad perception of the United States' "threat to world peace" -- reinforces my own: if one considers America Public Enemy #1, one better get to know that enemy and its intentions well.
You note: "I just cannot see what good it would do if they realised what their global image was like." Perhaps... though (to frame this in cynical terms) America certainly wants to exert its influence in world affairs, and to do that well it needs to succeed diplomatically; being more aware of the scope and kinds of opposition it faces -- the bodies who might kick back under threat of the elephant's weight -- is a matter of self-interest, which might, given the proper political leveraging, become a matter of mutual interest with others in the global bed.