What does it mean to write, speak, and listen in a professional manner, why it is so important, and how can your communication affect your career?

Writing, speaking, and listening in a professional manner means using formal laguage when trying to express yourself, giving examples or clear and concise explanations of your positions or thoughts, and being very attentive in choices in writing and speaking.
Your communication could affect your career because, often times, your speaking skills represent who you are, and the clarity of your responses or reactions to things may tell others the different ways you're dealing with things.

Hope this helps!