.MOV files not opening on Mac

Any ideas about how I can open these files? I lost a lot of videos of my kids.

Here's the screen shot:

Unfortunately, there are several potential issues here—only some of which may offer a recovery solution.

1. The MOV (QuickTime) file container is generic in nature. This means it may contain any form of compressed data that was compatible with the system on which it was originally created. If you upgraded from a pre-Maverics macOS, then the files may have been encoded with "legacy" compression data which is no longer natively playback compatible with El Capitan. If that is the case, then try playing the MOV files using a third-party media player like the free VLC player app. If the files open and play normally, then you can either continie to use this player to play the files or use an FFmpeg-based app like the free HandBrake app to convert the files to QTX playback compatible file formats.
2. Another possibility is that you transferred the files improperly to the "memory stick." There are several potential problems here. Your original files may have been wrapped in an MOV container that triggers a "security" issue on the updated system. Such files are normally playable in VLC and can usually be saved by simply copying the encoded data to a new MOV file container.
3. Another possibility is that your original files were encoded using a low-efficiency compression format like DV which produces very large files. Unfortunately, most "memory sticks" are initially formatted for MS-DOS (FAT) use which limits the file size drastically. In this case, if your original MOV files exceed the formatted file size limit of your "memory stick," then the files actually stored on the intermediate device would become corrupted since they would become incomplete/have no EOF (End of file) marker. These files cannot be recovered. (You can check the "memory stick" format by plugging the stick into your system, opening the Disk Utility app, selecting the memory stick, and checking the current physical volume format.)
4. Another transfer possibility for some users is the creation/storage of "Reference" files on intermediate devices. In this case, the new files don't actually contain any compressed data themselves but rather "point to" the data still stored in the original files. Unfortunately, since you "wiped" the original drive and QTX no longer supports such files, the data would again not be recoverable.

There may be other possibilities here. Normally I would recommend you first use the Finder "Info" window to check the size of you MOV files to see if they are ridiculously small and/or give any hint as to the compression format(s) contained in the file. As previously mentioned, since a "memory stick" was used for intermediate storage, I would also recommend checking the volume formats which have related file size limitations.