Why had Esme jumped of a cliff to try and end her life in twilight?

Because she did jump off a cliff. Earlier in New Moon, Jacob talked about cliff diving with Bella, and as that point in the book Bella was being reckless and wanted to try a lot of potentially dangerous things. This was also during the time when Edward was gone, and during that time Bella discovered that she could remember Edward's voice the most when he was angry, so while she was doing all these dangerous things she could hear Edward's voice subconciously telling her not to, and she liked that. So later she ends up jumping off the cliff, and that's because she figured that it was the most dangerous thing she could do, and would end up hearing hearing Edward's voice more than she had before.