And this isn't just making presentations and writing documentation. It's making effective use of your time in standups. It's conversations about your future relationship with your employer. It's knowing how to ask the right questions to weed out bullshit when digging through other technical documentation respectfully. It's taking a bold new idea that entered your head and figuring out which way would be the best to envangelize it at your organization.
I think in an ideal case, a master's degree in a computing discipline hits the intersection between intense training on a subfield of computing that can be difficult to break into (e.g. statistical machine learning, robotics, formal methods, cryptography) and your demonstration of your "mastery" is taking a cutting edge concept in that subfield and demonstrating mastery through effective technical communication in the form of a project/thesis.
A master's degree isn't guaranteed to make you a "better" programmer, but I would really hope that it would make you much more familiar with the field, and also teach you how to take that familiarity and leverage it to become a more effective communicator.