Backing up an iPhone 6 too damaged to be able to be unlocked
Unfortunately, there are not many options available should the screen be shattered. Depending on how valuable the data on your phone is (photos, texts, etc). then you might want to check into having the screen repaired. 

An iPhone 6 screen can be repaired Out of Warranty by Apple for $106 + tax at a local Apple store, and there are also Service Providers (authorized) and third-parties which will also fix the screen at similar costs which may be closer to you. I'd recommend you look into that option, as 


you will be able to retrieve your data (assuming no further damage than the screen)
You won't have to get a new phone over the issue, since the old one would be fixed


Beyond a physical repair of the device, your options would be limited. I might also suggest reaching out to a local third-party service as they might be willing to hook up a spare screen to your phone for the purpose of unlocking and backing up the data, but not a repair, at a lower cost should the full repair be too expensive. Of course, note that any 3rd-party repairs done by a non-AASP and/or non-Apple would void the warranty, should you have one.