He showed up
It's a fallacy to treat mythology as a consistent narrative with a single continuity. Lots of stories interact and combine, and the fragments that have survived become what we call "mythology".
We can get a better handle on this by tracing the references in the Wikipedia article to ancient sources. In some sources, it seems that Leucophrys was deserted before Tenes arrived.
In his telling of the story, Pausanias doesn't mention Tenes being made king:
Rejected by him [Philonome] falsely accused him before her husband, saying that he had made love to her, and she had rejected him. Cycnus was deceived by the trick, placed Tennes with his sister in a chest and launched it out to sea.
The young people came safely to the island Leucophrys, and the island was given its present name from Tennes.
Pausanias, Description of Greece, 10.14.2-3, Tr. W.H.S. Jones and H.A. Ormerod, 1918
Pseudo-Appolodorus also fails to mention that there were other inhabitants on the island:
Cycnus believed her, and putting him and his sister in a chest he set them adrift on the sea. The chest was washed up on the island of Leucophrys, and Tenes landed and settled in the island, and called it Tenedos after himself.
Pseudo-Appolodorus, Epitome, 3.25, Tr. J. G. Fraser, Theoi Project, Online Classics Library
We don't see the mention of Tenes being made a king until the 1st Century BCE historian Diodorus Siculus. But he tells more than one story:
Tennes was a son of Cycnus, who had been king of Colonê in the Troad, and was a man who had gained renown because of his high achievements. Gathering together colonists and using as his base the mainland opposite to it, he seized an uninhabited island called Leucophrys; this island he portioned out in allotments among his followers, and he founded a city on it which he named Tenedos after himself. ....
But we must not omit to mention what the myths of the Tenedians have to tell about Tennes, the founder of the city. Cycnus his father, they say, giving credence to the unjust slanders of his wife, put his son Tennes in a chest and cast it into the sea; this chest was borne along by the waves and brought to shore on Tenedos, and since Tennes had been saved alive in this astonishing fashion by the providence of some one of the gods, he became king of the island...
Diodorus Siculus, The Library of History, Book V, 83, Tr. C. H. Oldfather
Here Diodorus is saying that Tenedos came from a colonizing expedition to Laucophrys, and the colonists were already led by Tenes. But he allows for the local legend which says the people of Leucophrys making Tenes the king was solely due to his remarkable survival floating in the chest, a sign of the gods' favor.