I don't get the antiAmericanism either. It seems like a lot of people I talk to LOVE America, but can't stand Americans. Go figure.
It's simply out of ignorance. These people who speak badly of us see what they think is America on television, or even what they think is America on holiday, but they fail to realise that America is a MASSIVE, very diverse country of over 300 million people. They hear what our government is doing, and assume we all support it. I've been blamed for George W. Bush being elected, and although I was 17 in 2000, and voted against him in 2004, it's somehow my fault. It's a very, very small percentage of people I've met in the UK who think this way. The vast majority are very friendly and may make a small joke, but definitely aren't trying to insult us. Friends will make jokes, such as me not adding an "S" onto the end of the word "math", but I will have a comeback for them. That's part of British culture, though - taking the piss out of each other.
Many will see the stereotypical Texan on TV, and think we all wear cowboy boots and cowboy hats, or carry guns. The problem is, we don't have the same type of diversity in the UK. You obviously have big cultural variations from the UK's individual countries, but compared to what you have in the US, the cultural variation seen in the UK is much smaller. Every single Brit I've met has been to Florida, and we all know Orlando is definitely not a good representation of the entire US. In fact, I don't think there's a single place in the US that is a good representation of the entire country. However, many do base what they've experienced in Orlando on the entire US.
I watched an episode of Come Dine With Me, I believe they were in the Belfast area, and a woman came out and said "I don't like Americans". She decided she hated every single one of us. Of course, there happened to be a lovely American woman on the show. She was incredibly nice, a good cook, a good host, etc, yet this woman hated her simply because she was American. I was appalled by her, as I'm sure everyone was, regardless of nationality. I'm sure if she had been Irish, the woman wouldn't have hated her at all. You just come across the random bad seed once in a while, but, fortunately, they're not in the majority.
You also see the stereotyping from our side. Many Americans think that the English accent (the posh London accent, which they assume all Brits have) automatically makes everything sound intelligent, but when you live in the UK, you know that's not true.