If you want to say 'Santa Claus doesn't exist', you can't write a formula ~E x. x=Santa Claus. Because then you are conceding in your logic that some object Santa Claus exists.
So there is some connection between the structure of sentences we can construct in our logic and metaphysical claims. We should then try to understand differences in metaphysical claims by finding differences in logics.
The author then tries to show this for the metaphysical claim, 'Statements are true or false independent of whether they can be verified.' Instead of arguing about what is truth, what does it mean for a proposition to be verified. We can instead as about the differences in the logic. Because to even talk about truth and existence we need a logic first.
So don't think there is a claim that choice of logic forces claims about real world existence. I think is the more subtle connection, that choice of logic forces a choice about rules of inference.
Real world claims would need to be based on introducing propositions (which are independent of the logic?) and then using rules of inference (which are dependent on the logic).