Maybe a set of pictures will help. This is my method.
I have a bunch of these for various sizes. I had Frame Destinations cut these for me pretty inexpensively. These are made out of buffered paper, so they are pretty inexpensive. As you can see in one of the photographs, there are "photo corners" at 4 corners, so these are re-usable. Prints stay flat, protected from handling, and looks decent. I use these for casual presentations.
I take my prints out of my folders, put them in these temporary mats, and hand them over to whomever wants to see them.
I would imagine this is matter of personal choice but to me, photographs look so much better with mats around them. As I said earlier, I really don't like bare hands touching my prints, so this is my default method. There are all kinds of boxes available at art stores, you could buy them if you want. I made one myself out of available materials. I usually take them out before handing them over, so what it is made out of doesn't matter to me.