I think the argument is both true in some ways, and flawed. I agree that it takes a man (or perhaps the higher apes) to form the thought "I am in pain", and that most mammals don't bother with this type of reflexive thinking.
The flaw in the argument is that the "I am in pain" thought isn't the painful bit.
According to the theory behind cognitive behavioral therapy, believing that you're suffering exacerbates the suffering (and is often the major component). They apply this to physical suffering, too.
I ended up reading this article about animal suffering by this Christian apologist called William Craig. Forgive the source, please.
He continues the argument here.
How decent do you think this argument is? I don't know where to look to evaluate the core claim, as I know very little neuroscience myself. I'm quite concerned about animal suffering, and choose to be vegetarian largely on the basis of that concern. How much should my decision on that be affected by this argument?
EDIT: David_Gerard wins by doing the basic Google search that I neglected. It seems that the argument is flawed. Particularly, animals apart from primates have pre-frontal cortexes.