This is a question that I always address when I teach history. It applies both to ancient civilizations and to the impact of agricultural improvements on our own more modern society.
The major reason to me that food is necessary for civilization (other than that food is necessary for any life at all) is that civilizations need surplus food so as to support all the people who perform the various tasks that make a people civilized.
In order to have a civilization, you need to have government and education, you need to have scribes and merchants. If you have no surplus food, if your farmers produce only enough food to feed themselves, how can you feed scribes and teachers and merchants? All you can have is farmers struggling to stay alive (or hunter gatherers doing the same).
So, the way I always teach this is that surplus food allows for people to do specialized jobs. These specializations allow civilization because these specialized jobs make complex civilizations possible.
When I hear the word civilization, I think, stability. When I read this statement - I immediately thought of the movement from nomadic tribes to stable peoples who no longer had to move in search of food. I would not consider nomads to be civilized. Stability was brought about by learning to cultivate the land and grow food.
Consider also the first area that was reportedly civilized - the "Fertile Crescent." This area was so named for it's fertility in food production as a result of the stable water supply and rich soil.
Finally when I think of "civilized" I think peaceful. Think about all the times in History when a peaceful people turned on each other because of lack of food. (Many stories from the Holocaust come to mind.) It is similar to the idea that crime rises with poverty.
I guess your question really focuses on the forgotten importance of basic resources such as food when it comes to civilisations. Unfortunately, no matter how inspiring a leader a civilisation has or how mighty and vast the territory that a civilisation possesses, unless they have a good stock of food that is able to feed them and keep them alive, it all comes to naught. Note how this is particularly important in times of conquest. Armies must ensure that there are adequate food lines to feed their troops, otherwise any military endeavours come to a sad and premature end. I like this statement as when we think about great civilisations we don't necessarily remember the "bare necessities" that are an absolutely vital foundation for any civilisations.