I believe it was meant to be humorous.
Student or Learner
I can only do one thing at a time. Right now I am breathing.
I've just come across the quote and it made me wonder about the implication here. What is meant to say by the second sentence?
"Breathing" is an indispensable thing so we can't do anything else.
The speaker is deprecating the other person's importance by implying that the other person has so little value that any effort at all would be too much to spare for a response.
I am not a teacher.