Difference between 我饿了 and 我很饿? Someone tried to explain the difference in this answer https://chinese.stackexchange.com/a/43/38 saying that 我饿了 means "I became hungry", but I don't see the difference because we don't say I became hungry in English, you are either hungry or you are not (I appreciate this could be a cultural difference). 

So can someone try to explain to me further the difference between the two.

Same with similar constructs like 我累了 and 我很累.
I think that this is more a product of the fact that 我饿了 is something we don't commonly explicitly express in English.  We say "I am hungry" and let context and tone of voice determine whether this is some new information (I am hungry now) or merely the expression of a continuing state of affairs (I have been and continue to be hungry).  I have become hungry (to stick with the present) may sound like an awkward translation because it is trying to make the exact meaning of the expression explicit.  If we were translating a novel (and not tying to explain the language) we would probably translate it as "I am hungry" and allow the nuance to be suggested by the context.