Open source makes it is easier for security experts to review the code and determine whether it meets its security claims. If it is closed source, then there is no guarantee that the code the experts review is the same code that is used by the service. And also it is at the company's discretion whether to allow a security audit or not, and then which auditors to allow or exclude.

But if the whole of the client software is open source, then it also eliminates the need to trust binaries provided by the vendor (which can be MITM) because you can build the software yourself or use a version audited by an entity that you do trust.