That was a good start, backing up. But, indeed, it's only half of the total work.
How to restore depends on how you backed up. You forgot to tell that. The 2 extremes:
1. The backup program just copied the folders and files to another place. You can have a look with Explorer and copy/paste or drag and drop back to your hard disk.
2. The backup program wrote the files to some private format that only that particular program can read. Then install that program on your new hard disk and see what options it has to restore the backup.
With some bad luck, the program not only used some private format for the data but also encrypted it with a very secret key that only was stored on your old hard disk. Then those files are unrecoverable.
My hard drive got stuck. Happily I had backed up all my files, photos etc. It was under guarantee so I got a new hard drive. Now I just need to restore all my stuff.