I quite like their fence, but it does make yours look daft, whereas I'm sure it looked quite cute on its own.
Invite the around, ask them why they felt the need to put such a high fence up and talk to them about their plans. They may have already planted something (like clematis) and be happy to allow it to grow over the top then you can deal with it on your side. If not you can discuss attaching trellis to their fence/posts and put in some climbers.
I think you really need to talk to them.
Decide for sure first if you are happy with the height of their fence, because now is the time to object if you don't like the height of it, not once plants have been bought & planted.