Why is all this exterior trim rotting and how do I fix it?
The cause is simply nature.  You've got wood that's outside.  It gets wet, the wet stays in the wood, the wood rots.  This happens when wood is left outside in an area where water can get stuck and the wood wasn't properly installed (no caulk) and isn't properly maintained (not regularly painted/caulked).  

Much of the rot seems to start near places where wood was penetrated (nails, etc) which would make sense as an initial entry point for water.

The fact that it's on a corner near a downspout is probably a contributing factor.  You should watch this area of the house during heavy rain to see if the gutters/spout are leaking.

To repair - replace the wood, replace it with new wood, seal the seams between wood pieces with paintable exterior caulk.  Properly prime and paint the wood with exterior paint.  Apply new paint regularly as per manufacturer (the paint that is) guidelines.  Also check caulk regularly for potential need to replace.