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I remember when the term function was introduced in math - y=f(x) - the teacher was a substitution and still himself a student. He used the term function as if we already knew of what he was talking about. And the confusion was very big among us in the teacher training college. (Only months or years later we understood after many examples that the question was about to variables and their (inter)dependence.

Some years later when I started reading of harmony I learnt that tonic and dominant also are functions. And I remember how I asked my theory teacher Szandor Veress what is meant by function and he asked me back: "what is the function of a teacher?" and he answered himself: "the students", he said. And he asked: "what is the function of a policeman?" - "to rule the traffic."

I understood that also here must be a dependence of chords but it was not clear to me.

I also understood the term function of a tool or a job.

I only fully understood it last year when I read Ernst Kurth's book about counterpoint what functional theory is. Of course I knew the terms tonic and dominant etc. but I didn't find a clear explanation of the term function in the definition of "functional harmony".

People always use terms and terms of concepts without having really a concept.
It's easy to say the function V-I is the function of the dominant to the tonic. But is this not just a tautology?

Many questions and answers in this SE deal about function of chords and degrees, but they all imply that the meaning of function is known.

Definition of Functional Harmony

Today I know the answer.

But I wonder how you would explain it to a curious student that I (still) am.

See also: Definition of Functional Harmony

In functional harmony, simultaneous notes are interpreted as chords and the analysis is based around how the chords relate to the overall key and the preceding and following chords.

The relationship any one chord has in the context around it (i.e., the key and other chords) is called the chord's function. Another way to think of it is, the function of a chord is the answer to the question "what does this chord do to the harmony?"

So in the key of C major, any G major chord or G7 would be said to have a dominant function in that context, especially if the next chord is a chord that has a tonic function (the C major chord, in this example) and even more so if the previous chord has a predominant or subdominant function (which could be II AKA V/V or IV or I6/4, etc.)

Quick reminder: all music theories, including functional harmony, are just ways to attempt to understand how and/or why the music has the effect it has on most listeners. It's just a model or tool, it's never the full story. So while it can be helpful to understand chord function and functional harmony in general, such understanding is just a small, cloudy window into one aspect of the music.

I suppose it works like this in my mind:

Example, there are chords Dm, G7, and Am...

I can place them into a key C major...

The (Riemann) functions are basically the chord root identities labelled with Roman numerals ii, V7, and vi, further I can give general labels of pre-dominant, dominant, and (I selected vi on purpose) a non-tonic chord which gets' shoe-horned into the Riemann function system as another pre-dominant. If the Am had been C we would have the tonic and very neatly fulfilled the expected Riemann functions...

Riemann function is also described as a flow of events: pre-dominant, to dominant, to tonic.

Depending on the rhythmic phrasing I could say the harmony forms a deceptive cadence.

Normally, that is the end of the Riemann analysis story.

But, as I understand the question, the issue is not about the labels, but rather "what is the purpose fulfilled by the functions?"

So, in the Riemann sense: "what is the function of harmony?" answer "to lead to and form a cadence."

Notice that I specify cadence and not chord progression.

If we take something like V6 IV6 iii6 ii6 I6/4 V7 I, the initial chords in the progression don't fulfill Riemann function, but the ending does. To me, the essential functional part is the cadential part. Something similar could be said about sequential harmony where the sequential passage is a shifting of the tonal center - indeed, pivot chords get two labels - after the sequence a cadential passage confirms tonality at which point function becomes clear.

Whether any of this is satisfying intellectually or musically is another matter. My attitude about functional harmony - and harmony textbooks - changed a lot after I read Gjerdingen's Music in the Galant Style which turned me on to figured bass and solfege. Why should Riemann function trump rhythm and metrical function, or the function of consonance/dissonance and stability/unstability? All of which get's short shrift in most textbooks compared to Roman numeral analysis. "What's the function of a chord or harmony?" Answer: "to express yourself musically!" "Why does some music lead to cadences?" Answer: "because that is the convention is those styles!"

"Function" means "role" or "responsibility"; it's closer to the ordinary meaning, different from the specialized math term which means "mapping between two sets".

Western music theory works by identifying and classifying certain recurring patterns in music, such as certain chord cadences.

When we say that some chord has a function, we are stating (the hypothesis) that the segment of music seems fits some well-known pattern, and the chord is an element in that pattern. It is functioning as a piece of that pattern. Each piece of the pattern has a "job", and so if we know which piece of which pattern that harmony is, we can say that it's doing that job.

Sometimes there are ambiguities: more than one pattern applies to a segment of music. Sometimes that is deliberate; patterns overlap, so that for instance the ending harmony of one pattern doubles as the start of the next one. So then the same chord might have multiple functions at the same time.