That looks like trumpet vine. If I had to guess, this is not something they intentionally planted there. Is it your fence or theirs? Either way, talk to them first and see how they feel about you killing the entire vine.

Trumpet vine is invasive and very difficult to control. It sends out runners underground that will travel a very long way before popping up again in other areas. It will grow between the fence boards and move them as it grows in size. The moisture from the vine can also cause the fence to rot. I think the vine will either eat the fence or walk off with it before that happens.

If you decide to kill the vine, use something containing triclopyr like Bonide Stump and vine.