I just downloaded it and tried it out and it worked out for me. I have a Samsung Galaxy S3 (not sure how it'd work out on other phones).
I opened the PhotoStamper app. There are three tabs at the top. I imported the photo in the first tab (the one with the folder with a check mark in the middle). Then, I went to the second tab (the box with the arrows coming out of the corners) and played around with the size and position of the time stamp. Finally, went to the third tab (with the check mark) and clicked "Start stamping images". There is an option to change storage location, but I left it as the default. To see the time stamped image, I went to the Gallery app and searched for the folder named "PhotoStamper". My photo with the time stamp was there.
My guess for where things went wrong is probably that last tab. Double check you clicked the "Start stamping images" button and double check where the photos are being stored.
It seems as if PhotoStamper is not automatic (at least not the free version I downloaded), so you do have select which photos you want to add a stamp to. But, all of the time stamped photos end up in a separate folder which should make it easy to find when you are submitting images.