Translation:I don't watch news that much.
When あまり is used in conjuntion with a negative verb, 見ません, it means "quite/very." So in this sentence, it would be translated as "I really don't watch the news very often..."
(the trailing off signifies the way japanese often don't say what they actually mean due to cultural and social standards. In this question, the speaker is trying to say they don't really like watching the news without committing some social faux-pas, like if their boss asked if they watched the news segment last night.)
here's a link to a better explanation.