Nice question. There are several possible answers.
David Lewis maintains that counterfactual conditionals are statements about possible worlds, and that these possible worlds do exist. This position is an outlier - I don't know any other major figure who holds it - but it has the merit of being straightforwardly compatible with the idea of truth as correspondence.
Others go along with the possible world understanding of the semantics of counterfactuals but regard possible worlds as something other than real. They might be useful fictions. Or they might be permutations of properties or circumstances that are real. What makes a counterfactual conditional true relates to what similarities there are between a possible world and the actual world and how 'close' that possible world is. Robert Stalnaker is an example of an advocate of this position.
Some have claimed that counterfactuals are a species of strict conditional. They are implicitly modal, with the added complication of context dependence. The question of what makes them true then boils down to the general question of the epistemology of modality. In practice this still is often expressed as truth in accessible possible worlds with appropriate restrictions on the accessibility relation. This kind of position has been held by Anthony Gillies and Ken Warmbrōd.
Another option is to take a purely inferential approach to conditionals and hold that their truth consists in the fact that the consequent follows from the antecedent in some way, and this 'following from' is grounded in laws of nature or other established universals. This position was held by Nelson Goodman.
Some take the view that conditionals don't actually have truth conditions at all and should be evaluated instead according to assertability conditions. This is sometimes called the suppositional theory, or sometimes the "no truth value" theory of conditionals. Roughly speaking, the idea is that when you speak of the truth of "if A then B" you are really speaking of the truth of B within the supposed context of A. It is B that has a truth value, not the conditional as a whole. The conditional has an assertability condition given by whether it is probable that B given A. This approach has been defended by Ernest Adams and Dorothy Edgington.
This is not an exhaustive list. As you can see, it's a complex subject.