How does the story came to a happy end?

The story ends happily because the Robinson sons who want to leave the island do so, and the parents and other sons get to stay and live out their lives there—everyone gets what they want.
The Robinsons spend more than a decade on the island. After many years, they're joined by another shipwrecked person named Emily who they treat as a daughter. She and Fritz fall in love, and he plans to ask her to marry him with his parents's blessing.
The Robinson parents and two of their sons want to spend the rest of their lives on the island. Two of the boys want to go home again and see what they've missed. When a ship comes to take Emily back to England, they decide to go with her. The parents decide to stay. The island was discovered and isn't unknown anymore. They're hoping to build a happy society there with their remaining years.