Connect your S5 to your computer via USB. If anything pops up on the computer prompting you to install something (which is a typical Verizon thing to do), just exit out of that popup without accepting anything.
On your phone, swipe down the notification shade and tap on the USB Connection notification. Make sure it's set to MTP. On your computer, open Windows Explorer (assuming you have a PC), and look for S5 Mini as a device listed on the left column. Click on it, and you should be able to explore its folders. The phone's lockscreen has to be unlocked before the computer can get access. You can use Windows Explorer to drag and drop files from the computer to the phone.
Keep in mind that the S5 cannot connect as a "USB Disk Drive" or "USB Mass Storage," since that is an outdated form of connection which was phased out around the time of Ice Cream Sandwich.