I have been trying to explain the words 'have' and 'been' to someone who has learnt English as a second language. In trying to explain them, I've realised I don't really understand the meaning or function of the words individually, myself. Having only ever spoken English, I just know that a whole sentence using the words will mean something but I am struggling to define the words seperately. For example, 'I have been sent a letter.' What do the words 'have' and 'been' actually mean in this sentence?
These are auxilliary verbs, and as such they do not mean anything by definition. You cannot explain the meaning because there is none. They serve a grammatical function (perfect aspect, passive voice), not a semantic one.
An auxiliary verb is a verb that adds functional or grammatical meaning to the clause in which it appears—for example, to express tense, aspect, modality, voice, emphasis, etc. Auxiliary verbs usually accompany a main verb. The main verb provides the main semantic content of the clause. An example is the verb have in the sentence I have finished my dinner. Here, the main verb is finish, and the auxiliary have helps to express the perfect aspect. Some sentences contain a chain of two or more auxiliary verbs.
Auxilliary verbs exist in a great many languages, not just English. Chances are, your student actually has some in his mother tongue. If you know (or can look up) what they are, you'll find it rather easy to explain the situation to him.
The construction "have been ________" generally indicates the present perfect continuous, and is used to denote an action that has been happening for some amount of time until the present moment, e.g. "I have been studying for my chemistry exam." In your example, the rhetorical sense is muddier, since you were not continuously in the act of receiving a letter. "Was" seems preferable, but "have been" probably shades in a sense of "recently," drawn from the fact that the present perfect continuous usually/often describes recent actions.
There you have found a task that is actually rather difficult and I would never try to explain the meaning of "have" and "been" in a passive like "I have been sent". Either you take the view of John Lawler or you have to write a book about Latin conjugation, the change of this conjugation system in German and the second change of the Germanic system in English. No learner will understand it as such things are the basis of a university study in Latin, German and English.
What a learner needs is an useful conjugation table and an understanding of the system.
The English passive is of a baffling simplicity - the forms of to be + base form 3 (past participle). So you get
Tenses I (only the he-forms for the sake of simplicity)
- is, was, will be, would be + done
- has been, had been, will have been, would have been + done
The important thing for a learner actually is that he has nothing to learn. He knows already the forms of to be and to add the third base form is a piece of cake. But he has to know that
1 English perfect tenses are formed with forms of to have (and not as in German with to have and/or to be).
2 that after will/would an infinitive follows, either infinitve present or infinitive perfect.