There are many ways to do this. Owsn's suggestion is a good one if the backlight is out.
Have you ever connected your iPhone to a Mac computer or a Windows computer running the latest version of iTunes? This is one solution to back it up to the Photos app on a Mac computer. This will be hard if you've never linked the devices before because you have to tap on 'Trust' on your phone to allow the connection. If the backlight is out, and the digitizer is still operational, then this is still possible to do with a flashlight as Owsn has suggested.
Are you sure that the photos weren't backed up? Have you logged onto iCloud on a web browser to check?
Have you backed up your iPhone to iCloud recently? Could that backup possibly contain those photos and you could just restore your old iPhone from that backup?
If you can unlock your iPhone even with the screen broken, you could try a data recovery tool.
All that being said, the easiest solution is just to replace the screen. Not sure what model you have but replacement screens are fairly inexpensive on Amazon. Just be sure to do some research on that particular screen to ensure that there aren't any components that may be incompatible. Apple doesn't make it easy for us to be able to repair our own devices.
Hope this helps, cheers!
