Hi Lena,
We "keep things professional (at work)" when we communicate or deal with colleagues in an objective, factual and professional manner. We also refrain from bringing in our personal emotions, vested interests and personal disputes with certain colleagues which may cloud our judgment, action or choice of words. Someone who plays politics in the office does not "keep things professional at work".
Therefore, in your example, the note asks the reader not to be subjective and bring in personal feelings when answering Mr. Warbuckle's questions.