Why video is hard to recover
A common way to recover files is to find a file start, and then assume that the file data is sequential. For may items such as photos this works. On a FAT 32 memory chip, the address of each fragment of a file is stored in the File Allocation Table (FAT). When a file is deleted, so is the allocation data in the FAT. This means there is no way to 'undelete' the file because the order has been lost.
Why are GoPro Hero files stored in many fragments
The reason for this is that camera records both low and high resolution video at the same time. The data stream of clusters on the memory chip could look a bit like below
<high> <high> <high> <low> <high> <audio> <low> <jpeg> <high>
Each element is logically written into it's own file. Thus on a working chip, you will just see the files. However, remove the FAT (such as when deleted) and all you have is a series of clusters with data from one of the above streams. The length of each run of clusters is different for every video, and so is the sequence. Hence, finding the file start and hoping for sequential data does not work.
How does Gopro Recovery solve this problem?
The problem to solve is a bit like having a jigsaw puzzle and all the pieces upside down. Some pieces you can pick first, eg 4 corners, and the the edge pieces. The rest would be a bit of a guess - though one can assume that typically, but not aways, the pieces might still be in the correct area of the memory chip. Selecting these unmarked fragments in the correct order is where years of development has been spent to get the best possible results. The GPR program analyses each cluster of data and then reassembles each video using the correct clusters. This results in the original playable files. Thus the files are recovered, not repaired. Many reconstructed files actually have several hundred fragments and thousands of clusters.
On some chips, there may have been damage, making reconstruction impossble - ie the original data no longer exists. For these devices, the forensic version has some extra tools. Clusters are scanned but the meta data is then created from scratch. It means that small fragments of video can be viewed. The downside is that sometimes many small files can be created, instead of a single large file. The positive side is that even small fragments can be useful. At no time is any video data changed, so it remains forensically secure.