I've written on this topic before. The Transcendental Argument is an argument for the existence of God, though I would say that it is really more an argument against materialism. Materialism is a basic tenant of atheism, and if you dispose of materialism, atheism cannot stand. Thus its opposite, theism, must be true.
Before one can understand how the argument works, one first needs to understand what materialism is. Simply put, it is the belief that material (matter and energy) is all there as, and anything else doesn't really exist, but is merely an illusion in our minds. This has significant ramifications, for not only does it reject the belief in God, or in souls and spirits, but it also rejects the reality of morality, thoughts and even life. Life, after all, is merely a physical phenomenon, and thoughts are just electrical impulses in your brain. This is the philosophy of materialism.
According to the Transcendental Argument, there are things which we can prove exist which transcend material (hence the name). Most of the time, when this argument is used, the transcendental that is used is morality. However, proving the existence of objective morality is difficult (though I would say possible) and therefore this isn't always the best way to present the argument (sometimes it is though, because you can really engage the emotions of people).
Another one used is logic. Atheists love logic. Indeed, one may say that they worship logic and reason (Oh my science!). Because of this, using logic is a very good method of dismantling atheism in particular, since Atheism is so dependant upon claiming to be "the most logical stance".
However, I think a truly powerful transcendental to use, if all you intend to do is to deconstruct materialism, is thinking.
The Absolute Existence of Human Thoughts
As I stated before, materialism claims that material is all there is (hence the name). A necessary corollary to this is that our thoughts and consciousness are merely illusions: they are only electrical impulses in the brain. However, I would argue that an honest examination of human history and society demonstrates that this philosophy cannot hold water.
Question: have you ever heard of someone being run over by a unicorn? Of course not, because unicorns don't exist. This is the basic premise of my argument: Things which don't exist cannot affect things which do exist. Now I am in agreement with materialism that material exists. However, if materialism is to be true, then all events in human history can be explained purely from a physical level, without referencing people's ideas, philosophies, and beliefs.
However, this cannot be done. We can come up with some very simple examples like the notion of the atomic bomb started with an idea. It does not stand that the physical components of the human body and some electrical impulses are sufficient to explain the ability to break apart subatomic forces.
We can come up with something, though, which is a little more provable. I think the best example is racial segregation. Why is it that this group of humans (thought of in terms of physical bodies according to materialism) are in different physical conditions that this other group of humans? The true answer to this is racial prejudice. But racial prejudice is an idea. However, this idea must exist because it has physical consequence.
Indeed, any concept of a choice based off of a belief falls into this same category. Though my decision to have one piece of pie over another can be attributed to chemical impulses, my decision to place one DVD into a DVD player over another cannot.
OK, So Here's the Argument
OK, so here's the arguemnt:
P1: Material is real
P2: If materialism is true, then thoughts are not real
P3: Things which are not real cannot affect things that are real
P4: Thoughts affect material things.
C1: Thoughts are real (P1, P3, P4)
C2: Materialism is false (P2, C1)