A glib answer would be "because it's a huge continent"? In fairness, I think this is actually a pretty decent article about a complex topic, but I can't help but think it's the wrong question to ask.
If you combine just Mayan, Nahuatl, Mixtec, and Zapotec you're already in the millions.
I also think that the question suffers by failing to compare the range of languages found in other regions. The geographic distribution of the Indo-European language facility is comparable to Africa and contains around 500 known languages, though it is likely that at least as many were extinguished through various means (peaceful and not) over the centuries. Linguistic diversity is the norm, not the exception, though large-scale political consolidation tends to mask this.
Why is that an insightful question? I wouldn't have expected a strong correlation, and from the article, there doesn't appear to be one.
At the risk of being too dismissive, I thought the reasons for linguistic diversity were already well known (that is, physical and political separation and a lack of trade/communication).