Instructions
Connect the PSP to a computer, either using a USB cable or by using the Memory Stick directly. Check for the "Video" folder. Even though accidentally removing it won't delete the corresponding option from PSP's system, you won't be able to download, save and access videos until you create the folder again.
Make sure your videos are downloaded and stored in the correct file when using the PSP's Web browser. Try to download a video to verify it is saved in the correct destination. If the destination is other than the "Video" file, change the default destination. Retrieve your previously inaccessible videos from where they are stored in the wrong folder of your Memory Stick.
Check if you have activated the option of showing non-supported files. You may have downloaded video types that are not supported by the PSP and would not typically be shown as viewable.
Verify your files are in the MP4 format, which is supported by the PSP. Otherwise, a gray icon of torn film will remind you of your files' inaccessibility. Use the online Media Converter to convert your files.
Check if the videos have any added content, like subtitles or visual effects, that you used to enhance the video but later saved them over the initial file. Even if the video file was a supported format, it was possibly saved under as a different file type after the alterations. Convert them once again and you will be able to access them from your PSP.