The continuum hypothesis is known to be independent (neither provable nor disprovable) within the ZFC axioms. But as I understand it, mathematical realists (e.g. Platonists) believe that there is a single "correct" model of mathematics that corresponds to the real world, and therefore that every well-formed mathematical proposition is either "actually true" or "actually false", regardless of whether it can be proven in any particular axiom system.
The continuum hypothesis is perhaps the simplest and most intuitive claim known to be undecidable within ZFC. Do most mathematical realists believe it to be true or false?
The answer to your question may be geography-dependent. While on the West side of the Atlantic the dominant view is that CH is "probably false", on its East side and specifically in France, the opinions are heavily influenced by those of the Realist and Platonist Alain Connes who is convinced CH is true, as mentioned in this publication and also this. Connes discusses CH in detail in his A triangle of thought.
You should realize that it is not just because one calls himself a realist that one necessarily believes that CH has a definite truth value in set theory. Joel David Hamkins calls himself a realist; however he is not a realist of a set-theoretic universe but rather of a set-theoretic multiverse, where one slips effortlessly from a set-theoretic universe where CH is true to a set-theoretic universe where CH is false, at the click of a switch.