"used", in this context, is the past participle of "use", and is used in the passive voice. "used when you are politely asking for something" is not a clause, and thus does not have a subject or main verb. It does contain the clause "you are politely asking for something", which has the subject "you" and main verb "are", but the phrase as a whole is not a clause. It is an adjectival phrase, and functions much like an adjective does. For instance, if you see a word followed by the word "formal" in the dictionary, "formal" is an adjective that describes the word. Similarly, if you see a word followed by "used when you are politely asking for something", that is an adjectival phrase that describes the word.
Jack O'Flaherty's answer says "The words this word is are omitted because they will be the same for every word in the dictionary, and they are immediately understood by native speakers." but I think that it is more that there is no need to provide a subject and verb for adjectives and adjectival phrases. There is no need to say "This word is formal" rather than just putting "formal". Providing a noun (and note that all words are nouns, in the mention rather than use sense) followed by adjectives and/or adjectival phrases is understood to be describing it. Following a noun with other nouns is understood to mean that the first noun is an instance of those other nouns. If I post a picture of a car with a caption that says "my car", it will be understood that this is my car. I don't have to say "This is my car".