[Caveat: this is speculation; I don't study this myself. Second caveat: I am neither saying that it's scientifically okay to believe in creationism, nor that people who do are stupid; this post is about explaining, not justifying, the cognitive heuristics we use that make evolution so difficult to intuitively grasp].

Anyway...

Occam's Razor is a reasoning heuristic that says, roughly, that if two hypotheses both explain the data fairly well, the simpler is likely to be better. Simpler hypotheses, generally formalized as those with fewer free parameters, don't "overfit" the data too much and thus generalize to new data better. Simpler models are also better because they make a strong predictions. Such models are therefore falsifiable (one can easily find something they don't predict, and see if it is true) and, in probabilistic terms, put a lot of the "probability mass" or "likelihood" on a few specific phenomena. Thus, when such a specific phenomenon does occur, simpler models explain it better than a more complex theory, which spread the probability mass over more possibilities. In other words, a model with many free parameters -- a complicated one -- will be compatible with many different types of data if you just tweak the parameters. This is bad because it then doesn't "explain" much of anything, since anything is consistent with it.

When it comes to evolution and creationism, I think that scientists and laypeople often make exactly the opposite judgments about which hypothesis is simple and which is complex; therefore their invokation of Occam's Razor results in opposite conclusions. For the scientist, the "God" hypothesis (um, I mean, "Intelligent Designer") is almost the prototypical example of a hypothesis that is so complex it's worthless scientifically. You can literally explain anything by invoking God (and if you can't, you just say "God works in mysterious ways" and feel like you've explained it), and thus God scientifically explains nothing. [I feel constrained to point out that God is perfectly fine in a religious or spiritual context where you're not seeking to explain the world scientifically!] This is why ID is not approved by scientists; not because it's wrong, but because it's not falsifiable -- the hypothesis of an Intelligent Designer is consistent with any data whatsoever, and thus as theories go ... well, it isn't one, really.

But if you look at "simplicity" in terms of something like number of free parameters, you can see why a naive view would favor ID over evolution. On a superficial inspection, the ID hypothesis seems like it really has only one free parameter (God/ID exists, or not); this is the essence of a simple hypothesis. By contrast, evolution is complicated - though the basic idea of natural selection is fairly straightforward, even that is more complicated than a binary choice, and there are many interesting and complicated phenomena arising in the application of basic evolutionary theory (simpatric vs. allopatric speciation, the role of migration and bottlenecks, asexual vs sexual reproduction, different mating styles, recessive genes, junk DNA, environmental and hormonal affects on genes, accumulated effects over time, group selection, canalization, etc). The layperson either vaguely knows about all of this or else tries to imagine how you could get something as complicated as a human out of "random accidents" and concludes that you could only do so if the world was just one specific way (i.e. if you set many free parameters just exactly one way). Thus they conclude that it's therefore an exceedingly complex hypothesis, and by Occam's Razor one should favor the "simpler" ID hypothesis. And then when they hear scientists not only believe this apparently unbelievable thing, but refuse to consider ID as a scientific alternative, they logically conclude that it's all just competing dogma and you might as well teach both.

This is a logical train of reasoning on the layperson's part. (Doesn't mean it's true, but it's logical given what they know). The reason it doesn't work is twofold: (a) a misunderstanding of evolution as "randomness"; seeing it as a search over the space of possible organisms is both more accurate and more illuminating, I think; and (b) misunderstanding the "God" hypothesis as the simple one.

If I'm right that these are among the fundamental errors the layperson makes in reasoning about evolution, the the best way to reach the non-mendacious, intelligent creationist is by pointing out these flaws. I don't know if anybody has studied whether this hunch is correct, but it sure would be fascinating to find out what sorts of arguments work best, not just because it would help us argue effectively on a national level, but also because it would reveal interesting things about how people tend to use Occam's Razor in real-life problems.