How might we parse the phrase "put [something] out to board" to better understand its meaning?

The phrase is found in John Cheever’s The Swimmer:

He wondered if the Lindleys had sold their houses or gone away for the
summer and put them out to board.

I understand the phrase to basically mean rent out, so that he could have said: "…or gone away for the summer and rented them out".

Are there accepted usages of ‘put out‘ and ‘to board‘ that when coupled together yield the meaning ‘rent out‘?

It seems Cheever made a mistake here. For ‘board‘ means "to provide with regular meals and often also lodging usually for compensation". But the sentence quoted states that, in the instance of boarding, the owners of the house had left. In that case, it seems they wouldn’t be able to provide any meals.

And as for ‘put out‘, I can’t find anything in the dictionaries to suggest it has a history of meaning anything like ‘rent out‘.