What factors make Central America's biodiversity rich and unique?

Rainforests tend to have high biodiversity

Geological and geographical diversity drive the biological diversity.

The area is at the intersection of two continents and two oceans and the biota is influenced and contributed to by all four.

Because of the range of latitudes, altitudes and proximity to the oceans which act as heat sinks and water sources the area has a huge range of climactic associations ranging from freezing and dry to sweltering hot and inundated by rain.  The more climates you have the more niches there are to fill and the more species you will encounter.