This is my stab at it:
Imagine the universe just a million years after it was born. A batch of light travels for a year, covering one light-year. At that time, the universe was about 1,000 times smaller than it is today. Thus, that one light-year has now stretched to become 1,000 light-years.
All the pieces add up to 78 billion-light-years. The light has not traveled that far, but the starting point of a photon reaching us today after traveling for 13.7 billion years is now 78 billion light-years away. That would be the radius of the universe, and twice that -- 156 billion light-years -- is the diameter. That's based on a view going 90 percent of the way back in time, so it might be slightly larger.
I was explaining it to our engineering director and he finds the explanation completely lame, I don't know how to make it clearer.