Welcome to Android Central! I don't know why Facebook does it this way, but do the following:
1. In Facebook, select the photo you want to download, tap the 3-vertical-dot menu button, and tap Save to Device.
2. Now use a file manager app like ES File Explorer to navigate to /storage/sdcard0/com.facebook.katana/fb_temp
3. The saved photo should be there.
I also notice a ".nomedia" file in that folder, which means your Gallery app won't be able to see it. You can use the file manager app to copy or move the photo somewhere else, like the Pictures folder.