My answer to this question also answers your question, I think. The key point: many set theorists don't think that these questions are matters of convention, but rather (to quote Drake), "in some sense sets do exist, as objects to be studied, and that set theory is just as much about fixed objects as is number theory."
Examples of this attitude can be found in Gödel's essay "What is Cantor's Continuum Problem?" and in the conclusion to Cohen's book Set Theory and the Continuum Hypothesis, also Drake's book Set Theory: an Introduction to Large Cardinals, from which the above quote is taken. The first sentence of Schindler's book Set Theory: Exploring Independence and Truth expresses it succinctly: "Set theory aims at proving interesting true statements about the mathematical universe."
Suppose $X$ is a statement in the language of Peano arithmetic that cannot be proved in PA, but can be proved in ZFC. (Examples: the Paris-Harrington theorem, or Goodstein's theorem, or the assertion that PA is consistent.) Most number theorists are happy to say that $X$ is "really true", not that its truth is a matter of convention.
Analogously, many set theorists think that $2^{\aleph_0}$ has an "actual value". Yes, the axiom system ZFC can't determine what it is, but that's evidence of the weakness of this axiom system. We need more axioms that are "true about the universe of actual sets", or so the thinking goes.
It's also worth noting that the axioms for ZFC weren't written down at random, but flow from an intuitive picture, known as the cumulative hierarchy. Perhaps this intuition can be pushed further, suggesting new axioms that will settle CH. Or so it is hoped.
I should note finally that Platonism is by no means universal among set theorists, or even mathematicians at large. Many would agree that the adoption of CH or an alternative is a matter of convention, or personal preference, or whatever, but not a question of "truth". For a more extensive discussion, see the entries Platonism in the Philosophy of Mathematics and The Continuum Hypothesisin the Stanford Encyclopedia of Philosophy, and the panel discussion "Does Mathematics Need New Axioms?" in the Bulletin of Symbolic Logic, 6(4) (Dec., 2000), pp. 401-446, with participants Solomon Feferman, Harvey M. Friedman, Penelope Maddy and John R. Steel.