First, we need to understand that neurons have special proteins embedded in their membranes called ion channels. These allow dissolved ions such as sodium, potassium, and calcium to pass from one side of the membrane to the other. Neurons use these channels to control their electrical activity.
Peptide neurotoxins such as the one produced by funnel spiders (robustoxin) work by having a shape and charge configuration that is complementary to ion channels, and thus they will bind to these channels and affect the way they operate (often by simply blocking the channel). Robustoxin seems to operate by removing sodium-channel inactivation.
There is a bit more information about the primary toxin on the Wikipedia page for delta atracotoxin. The intended target of this toxin seems to be insects, where it interferes with potassium and calcium channels. In mammals it interferes with sodium channels, but it seems to be particularly potent in primates. Why are some species affected more than others?
Not all ion channels are built the same. From species to species, we see a huge array of genetic differences that affect both the shape and functional properties of these proteins. Ion channels which happen to have a shape and charge configuration that is highly complementary to robustoxin will adhere to it more firmly, and thus be more affected by it.
So did we just get unlucky and happen to have sodium channels that perfectly fit robustoxin? That seems to be the most likely answer, given that Australia probably has no native spider-consuming primates. Perhaps someone else can suggest other arachnid predators that might be an intended target for robustoxin?