In Lutheran liturgy the "Greeting" (The Lord be with you / and also with you) is chanted 3 times: before the Collect, at the Preface, before the dismissal. But what does it mean? What is the logic behind it?

Several explanations have been offered, and there is room for interpretation, but I think we would do well to consider the "greeting" specifically Eucharistic in nature. To think of it not merely as a wish but as performative speech: speech which accomplishes what it says. This would be a fitting interpretation because in liturgy the Lord (Jesus) factually comes to his Bride in intimate communion, and is "with her" in as real a way as ever could be (even if under the forms of bread and wine).

And so what we say: "The Lord be with you. And also with you," comes true, and is performed, as we take and eat the Lord's glorified flesh and blood here given.

What is the evidence? Almost all of Paul's epistles segue into the Liturgy of the Sacrament. We see this most clearly in 2 Corinthians 13:14 which is not simply a benediction, or "the apostolic benediction". But was likely the Preface which led into the Eucharistic portion of Corinthian liturgy. At least that is the way it was taken from earliest times if you look at ancient Eucharistic liturgies. Most, if not all, begin with that formula, or some variation thereof.

In western liturgy that once longer formula was reduced simply to "The Lord be with you." But the function is the same; a piece of performative speech that comes true in the eating and drinking.