What are transport planes good for in Hearts of Iron 4?
Transport Planes are used to stage paradrops with Paratrooper divisions.
They work similar to naval invasions, just their range is limited, you need air superiority to execute them, they are much faster and instead of convoys, you need enough transport planes in the local air base to bring all your paratroopers to the target location. However, immediately after a drop Paratroopers have minimal organisation, so dropping them directly into combat is almost always fatal unless they completely overwhlem the enemy. So, drop them behind enemy lines for the best effect.
In addition to Dulkan's answer,
With the Waking The Tiger DLC you can use them to do the Air Supply mission to supply land troops with transport planes. I believe you do require Air Superiority for this to work.