Literally, the king eats Madame de Sauve with his eyes. That’s an idiom meaning he stares at her, pleasingly, hungeringly, maybe even, to stay in the “edible” domain.
Depending on the context, this can often have heavy implications (maybe with the eyes is enough for now, and later on you know what I mean).
I’ll throw, adding to Gildas’ comment, that the idiom can be varied a lot by replacing manger. In fact, I wouldn’t be surprised if dévorer des yeux was more common.
The somewhat related idiom manger avec les yeux is fairly common, referring to how food’s aspect influences the consumer.
He is starring at her because he finds her attractive, sexy.
As Circeux mentioned, “dévorer” is more common now: “dévorer des yeux”.