The term "flattening of affect" is a term of art used by therapists to describe what happens to one's face if one is showing little or no emotion. This is the one instance in which the word "affect" is used as a noun instead of as a verb, so people not acquainted with the term are likely to find this confusing. The opposite of flat affect would be animated affect, meaning that a person's face is showing his or her emotions.
Often flatness of affect is a symptom of mental illness, for example, depression, and this is often used to help in the diagnosis of various mental illnesses. You may or may not have observed someone with a flat affect, but now that you know what this means, you might find it interesting to observe how different we are on the continuum of animation in our faces.