It's about cause and effect. An embarrassed silence happens BECAUSE people are embarrassed. If people are talking, and then one person says something awkward, everybody falls silent because they feel embarrassed and don't know what to say next. An 'embarrassing silence' CAUSES people to become embarrassed. If you are giving a lecture and you suddenly forget what you have to say, this results in a silence which embarrasses you.

"Embarrassed silence" and "embarrassing silence". The difference between these two phrases (both of which are grammatically correct) is in relation to the participants in the sentence. "Embarrassed" refers to one or other of the participants being embarrassed by what has gone before (who would depend on the context). An embarrassing silence is more distanced from the personal/individual and implies that the lack of something to say is awkward for all participants, but it does not imply anything about the preceding content. "her remark was followed by an embarrassed silence".