"Logic is the technique by which we add conviction to truth." ~ Jean de la Bruyere
It is not that logic is the only path to truth. But it is the only path to verifiable truth. Logic, however, is not the tool of every atheist. I've met some very illogical atheists in my day. Because often, what people (theist or atheist) call "logic" has little to do with the discipline itself.
The laws of logic are realistic or modeled after realistic principles. It should be possible for anyone to understand them. However, according to my own experience, few people actually do.
What's so hard about axioms, causality (cause and effect), premises, conclusions, validity, soundness, analytic agreement, synthetic agreement, subjective evidence, objective evidence, conditions, functions, truth values (factual truths / tautology), deduction, and induction?
Even without understanding the laws of logic, one can use logic. You just can't use it correctly.
They work because they are based on realistic principles that are evident in our external reality and are not just products of the mind. Usually referred to as "transcendental logic" via Kant. Logic that was just a mental projection would be useless in determining realistic aspects of the world because anyone who used logic would do so according to their own biases and beliefs, and everyone would be equally correct in the absence of a standard of synthetic agreement--that link between thought and the material.
Nothing that actually exists is technically outside the bounds of logic. That is like saying something is outside the bounds of physics. If something cannot be explain by logic, it is likely because it does not exist in the first place or is unknowable by us.
No claim for phenomena lying outside physics has ever been proven or even explained how that would be possible. Many things we once thought were, ended up not being so. The same with logic.
Our ignorance of the logic of a particular event or concept does not invalidate the logic itself. It just demonstrates what we already know which is we don't know everything.
