I believe it comes from the idea that in order to eat a pizza, you need to have one first. If you don't have pizza, you can't eat it. So, "have" became the same as "eat." Think of ordering in a restaurant; you wouldn't normally say, "I want to eat spaghetti," but rather, "I'll have spaghetti." The implied meaning is that by "having" it, you will also be "eating" it.

As jpellowe says , 'to have ' a pizaa rather means to order it and it implies having eaten it thereafter. To say 'I ate a pizza' emphasizes more on the act of eating ,the process itself,it could be at home,at a friend's , a restaurant or anywhere. 'I had a pizza' stresses the fact that you were in a restaurant and orderd it and of course ended up eating it.

I'd like to bring up another point that others have not yet mentioned. I think using "had" in this case can be context driven. "I had pizza" may give an implication like, "I had pizza for lunch today." In this case, you had it for a meal, which must mean you ate it. I agree with others that in general, if you say you had any food or drink, it would imply that you ate/drank it. However, if you ended your sentences differently, this implication may not apply. For example, "I had a pizza, but I dropped it." In this case, you would not think that the person ate the pizza. Again, using "had" leaves it up to context as to what it may or may not imply. Using "ate" leaves no doubt at all.

To be a hairsplitter, eat is always more exact, I had a chicken, I ate a chicken, see?

I had a pizza. I ate a pizza. Same meaning.