A transport plane is an original plane unit from the game, The Conquerors. It is called a transport plane because of its ability to garrison units. It can carry 6 units in total.
In the beginning of games, if players buy an airport and get a transport plane, they will either use it to raid enemies or use it to claim plants, due to it not damaging objects. In mid-late games, people use transport planes to raid player's bases with walls, or get to a certain place faster. A transport plane cannot carry a plane, a tank, nor a building.
When an enemy is far away from you, like on an island or opposite land, it is better to have transport planes than transport ships to get your units where they need to be in a dire emergency or just to get your army ready.
Transport planes are good if troops need to reach areas with higher terrain, especially if there is no ground access to it (Desert vs Grass etc). Even if it can be accessed with slopes etc, transport planes make troop movement to mountains much quicker.