The first thing to try, and usually the easiest, is to talk to the neighbor who owns the plants. It may be that he is not aware that there is a problem. Contacting him and showing him the problem, or pictures of the overgrown and damaged fence, may be all that is required in order to get him to take action and cut back or remove the plants. He should be willing to fix the fence as well, or to pay for its repair.
Property Owner's Rights
When a neighbor does not respond favorably to a polite request to remove the plants, the property owner should know her rights before taking further action. It is important to check what the neighbor's obligations are and what remedies the property owner can take before doing anything. Start with the local regulations, which often override state regulations. Check with the homeowner's association to see if there are rules that deal with such things in the neighborhood. Otherwise, check city, county and then state ordinances regarding nuisance plants and property encroachment.
In many areas a person who owns a tree or other plants that are encroaching on someone else's property is responsible for removing the offending plants and repairing any damage they have caused. He does not have to remove any plants that are on his property, but he must remove those that are growing from his yard onto someone else's land if requested to do so. Trees are sometimes handled differently, but vines and bushes are usually simply removed.
The person whose property is being damaged by a neighbor's plants has the right to remove all parts of the plant that have grown onto her property. The owner of the plants should be notified, but the property owner of the property that is being damaged is within her rights to remove any plants that are on her side of the boundary line. Usually the neighbor is responsible for paying for the removal of the plants and the repair of any damage they have caused, but it may take a court order to get him to pay. Document any damage with photographs and witnesses before taking any action, in case it becomes necessary to go to court.