the context of this sentence is convicts were transported from England to Australia in 1780:

"and most of the convicts had left their country for their country’s good. A few were never to see England again."

"left their country for their country's good": does it mean those convicts were doing it for the interest of their homeland? hardly true, as they were forced to Australia. or, does it mean "left their country for good"? again, there is a problem: the next sentence says the exactly the same thing.

so what is the exact meaning of the sentence then?
It means that it was for the benefit - the good - of the country. It doesn't say they were doing it altruistically or that they wanted to go. It just means that it was good for the country that they had to leave. I agree it does sound a little strange.
A more exact formulation:

had left their country->had been forced to leave their country

but when reading the word convicts, one realizes the true meaning, they certainly were forced to do it.