I'm not sure if this is the way it is supposed to work but units can only enter transport ships if the unit is inside a city and the transport ship is adjacent to the harbour tile. I was expecting units could be dropped off and picked up from beaches but picking up from a beach doesn't work. This means you must develop the second island with another harbour before you can pick up any unit you drop off there.

I've also had a problem in one city where the harbour was facing inland (not sure how to rotate the tiles or should these rotate automatically?) and I couldn't load units onto the transport ship.

Finally, transport ships travel way too slowly in the game and are annoying waiting for them to move their couple of squares.