Should I try to explain what the word yid means?
no, after pointing out that you are Jewish thank them for their proactive stance against anti Semitic references, and just say as a Yiddish speaking person yid is certainly not offensive to yourself, but you understand others may take it that way. In my experience non-frum people have a very hard time with the nitty gritty of anti semitism, frum lingo etc. (had that experience with a chasidic friend named luzer (eliezer) in a non-Jewish venue(but if you know that in English looser is derogatory who cares that you are referring to his Hebrew nickname etc). Keep it simple and be thankful, we are in galus after all