I've heard a number of analogies trying to explain it, but I've found all to be flawed, according to my monotheistic, Jewish mind. So I still don't get it.
A note on your conversation with
@Riders.
None of the five versions of Jesus in the NT ever says he's God. Instead, each of them says clearly at least once that he's not God.
Notwithstanding, in the 4th century CE there was formidable political pressure to make the central figure of Christian worship God. Unfortunately it was politically unacceptable to have two gods, Father and Son, because that would be polytheistic hence pagan.
A number of possibilities were rejected or implicitly excluded: God is NOT three gods. God is NOT a corporation with a board of three. God is NOT a partnership with three partners. God is NOT the sum of his parts.
Instead, the one God "exists as three persons and one substance". (I'm quoting the
Oxford Dictionary of the Christian Church.) The persons are the Father, Son and Ghost ─ and the Father and Son, Father and Ghost, and Son and Ghost are all distinct from each other. But although there's only one God, each of them
is God ─ and since God is not the sum of his parts, each is 100% of God.
Now 100%+100%+100%=300%= 3 gods, so there's a problem here. The problem is dealt with by declaring the Trinity doctrine to be "a mystery in the strict sense" (which distinguishes it from the church's more ordinary "mysteries").
This means that "it can neither be known by unaided human reason apart from revelation, nor cogently demonstrated by reason after it has been revealed" (
Oxford Dictionary of the Christian Church again).
Therefore, as a moment's thought reveals, the expression "mystery in the strict sense" is a synonym for "an incoherent notion" ─ "a nonsense" if you like. I assume they prefer "mystery in the strict sense" because it sounds better.
You'll recall from the NT that the Jesus of Mark had ordinary earthly parents and only became the son of God by adoption, just as King David had done; that the Jesuses of Matthew and of Luke didn't exist until the divine impregnation of Mary (so that they have God's Y-chromosome); and that only the Jesuses of the two gnostics, Paul and the author of John, pre-existed in heaven with God. The Trinity doctrine, as you can see, sides with the last two and against the first three.