Two ways.
If you press Print Screen, it saves it as a copy + paste item. Which means you have to hit Print Screen, then open up Microsoft Paint, go Edit -> Paste, and poof, it puts the image in paint.
There is another way, however, but it works ONLY while ingame on SFC3, it doesn't work on other programs. Press F12 and it will save a Screenshot in the SFC3/screenshots folder.