Reading through the available literature there appear to be three main theories for the cause of the decline (that I'd consider credible anyway): environmental degradation, warfare and disease, and climate change. Typically these are cited as a group of possibly complementary possible causes.
The main idea behind the environmental degradation theory was that Cahokia, as structured in 1200, consumed a tremendous amount of wood, and after a few hundred years of over-harvesting the area simply ran out.
Deforestation required longer walks for firewood. Charred remains show that Cahokians burned oak and hickory in the early years but used energy-poorer soft woods later, a sign of problems, Iseminger says. The stockade alone required as many as 20,000 poles.
I'm kind of skeptical of this as a sole or primary cause. Civilizations destroyed by their own hands this way tend to either be very isolated (like the Easter Islanders), or working very marginal agricultural land (like the Maya). It could certainly have contributed though.
As to warfare: there does appear to be more sign of defensive structures both at Cahokia, and at neighboring Mississippian towns, starting at around 1200. While this no doubt contributed, IMHO increased warfare tends to be a symptom of other problems (generally a sudden shortage of resources), rather than a root problem itself.
There has also been evidence found in remains at the site of endemic disease problems (and no evidence of provisions for sanitation). However, disease is a problem that tends to go hand-in-hand with large cities. IMHO it would be far more remarkable if they didn't find evidence of widespread disease.
Now for the Climate Change theory. One thing that does jump right out at one is the timing. It turns out that there was one other agriculture-based civilization that was wiped out of North America at seemingly the exact same time: The Greenland Vikings. They were a (somewhat) literate people, who coincidentally went into decline around 1200 and were last heard from in 1410. There is still some debate over their decline as well, but the preponderance of evidence points to the global cooling period known as the Little Ice Age.
Based on radiocarbon dating of roughly 150 samples of dead plant
material with roots intact, collected from beneath ice caps on Baffin
Island and Iceland, Miller et al. (2012) state that cold summers
and ice growth began abruptly between AD 1275 and 1300, followed by "a
substantial intensification" from 1430 to 1455 AD
The colder temperatures and shorter growing seasons would have made agriculture in marginal areas untenable. This is what records indicate happened to the Vikings of Greenland.
Now the staple crop of the Mississippian region was Maize. This is a crop ultimately of Central American origin which took thousands of years to evolve variants capable of being intensively harvested in temperate regions. While Cahokia may not have been at the extreme northern boundary of viable high-intensity maize production, it was certainly near it. A prolonged period of cold weather, which we know for a fact happened around then, would have made the populations Cahokia had at its peak simply unsustainable.
If you check around, this is a theory that pretty much every source mentions prominently. While I can't find anyone saying they believe it is the primary cause, the fact that everyone brings it up, many of them most prominently, is probably significant.
So while there are a variety of theories and explanations, the one that seems the most compelling (if one is forced to pick) as a primary cause would be the Little Ice Age.
(note: One particularly useful source I found was Cahokia Mounds: America's First City. If you're interested in this topic, you may consider picking up a copy)