There are two directions one might take to this. The first is to say that Logic is, in generality, the study of inference, rather than thought per se. One studies the relationship of premises to conclusions, and while this relationship might be usefully applied in considering the nature of thought, not all logics need be thought. For example, Godel's model of formal logic works over a language coded entirely in the syntax of natural numbers, which does not seem to be a reasonable representation of human thinking practice.
So one might think your problem simply is a question of using the mathematics of logic as a model of human inferential reasoning in practice. Meta-logic is just Meta-mathematics, and that seems to be not particularly bizarre (than mathematics is by itself, at any rate).
For the other direction, this concept of "representation" draws out something interesting about the notion of human thinking, which is that in as much as we seek to model the "thought processes" of other humans, it has a very strong semantic component. The idea of receiving a visual stimulus and perceiving the world seems to be of a different character to what we would want to attribute to them as holding this particular image of the world in thought.
One might say that you've got things the wrong way around. Logic, or more generally our Language, patterns of speech and word association, and Grammar, partly constitute "thinking". We don't actively represent the world in logic, but rather our linguistic/semantic limits determine the boundaries of how we "think".
That's because when we try to work out what we mean by "thinking", we've historically had to do so with reference to asking other minds questions and listening to their responses. Human cognitive capacities may well (in fact they almost certainly do) outstrip this perhaps limited notion of thought, but that's not a fault of logic, but rather how our capability to infer the possibilities of the inner minds of other people has been filtered through the channel of linguistic expression.
Now, with the advent of modern experimentation, this boundary between minds seems less linguistic, and the concept of "thought" seems like an artifact of folk psychology. Yet at the same time, we want to try to make sense of the cognitive individuation of areas of the human brain with reference to behaviours and reports of behaviours in order to demonstrate "what the bits of the brain do". This line of investigation goes into the burgeoning field of Cognitive science, and your question of "how is Cognitive Science possible?" might well result in some interesting philosophical discussions of its own.
In short then, have a mull over exactly what aspect of "logic" you're really interested in, and delve into one of the two avenues of discussion this draws out. One line's metatheory is very different from the other, though both are almost certainly going to be fascinating.
(Also, thanks for this question. I was halfway through an initial answer when I was led to Wittgenstein's Philosophical Investigations to sense-check something I was about to attribute to him. I normally hesitate to recommend it, but it does seem entirely appropriate here - Wittgenstein's musings very much stand in for this kind of consideration about the limits of language and logic, and his theraputic remedies are particularly useful.)