tl;dr Feedback isn't feedback unless it's Specific, Measurable, Realistic. These lead in to SMART (Specific Measurable Agreed-upon Realistic Time-based) goals that are actionable. Use this a prompt to talk to your boss, and find out what they want from you.
edit: What to do? Tell them this. You really appreciate feedback, and would like to have them put them in to SMART goals for you to address. Without them, you've got nothing to address. Depending on your relationship with the professor, you can also ask that they push back when given this style of feedback.
"You lack communication skills" is not feedback, it's slander. Just as we close questions for being too broad, or unclear as to what they are asking here, this person is having you chase a ghost. Worse yet, this "feedback" was given directly to your professor, instead of them working with you on this. It's not specific, measurable, or realistic. "Communication skills" are discipline unto its own.
Feedback surrounding such broad topics demand examples and clarity. In this case, if I was the boss's supervisor, I would say something like "Boss guy, I'd like to talk with you about some of the feedback you provided to Elena93. I heard that she was lacking in communication skills. This is concerning to me, and I'd like to know more details surrounding that." Then continue to ask questions and be curious as to why this low quality feedback was provided. Curiosity should be the approach you take with both your professor, and your boss.
It's a shame your professor didn't ask more questions when the boss approached them. They should be interested in protecting their students from this kind of near-useless commentary.