... Plugged in the charger cable, clicked on the "charging" notification, and then the options came up. Chose "file transfer", and viola! Back to normal. Using a Verizon Galaxy S5, locked to the carrier, of course. Problem solved.
I just ran into the problem last night (Aug 21, 2016) at a friend's house. I wanted some files, but neither of us had a USB drive. I had my data cable with me, so I just thought "Why not, I will just plug my phone in and toss the files on the SD card." but his computer would only show my phone as a phone in My Computer. It wouldn't let me explore it or browse files. We were pressed for time, so i didn't play much. I just assumed there was some driver issue that wasn't setup on his PC and we just left it.
Today (Aug 21, 2016) I went to backup all the pictures off my phone in the way I have done a hundred times before, using the data transfer cable plugged into a USB port on my desktop. I got the same result as I did at my friend's house. It brought up my phone in windows explorer, but wouldn't let me explore the files. It would just give an error and a blank screen when it was double clicked.
I tried the original solution in this thread of dialing #0808#, but my phone really dialed it and I got a generic Verizon call canned be completed. My phone must me somewhat locked down by Verizon (no surprise). So I went and manually explored the settings looking for an option that a recent system update might've changed. No luck. I finally noticed I had a notification that my phone was USB charging and to tap it for more info. I pulled down the notifications, tapped the USB charging notification and viola! I had an option to change it to transfer files among a few other options.
My guess is that a recent update changed it. I haven't transferred files via cable in a while, so it is very possible that it was changed some time ago and I just now noticed.
Just felt it was my civic duty to chime in and document my problem / solution here too since I had an account. Thanks!
(My phone is a Samsung Galaxy S5 running system version G900VVRU2DPF4 as of Jun 25, 2016)