You need to develop the island using beans. When you arrive click on the bottom right button and than click on developed Isles.
Pick Isles Avue and develop it 3 times to get 18 slots to leave your eggs. Clicking on one of the signs and a message will pop up.
Click on that and it will take your pokemon box. Add 6 eggs and than click on the bottom Left bottom. Repeat until you have 18 eggs.
You can leave beans in the box at bottom. It helps speed the hatching process.
Check on the eggs later to see how close they are to hatching. If an ! is above the egg those are done. Touch the egg and it will hatch. If it's very close to hatching I take those out and replace them with new eggs. Leave the island and take the ones out that were close to hatching. Walk a small amount and those ones usually hatch.