How can one approach believing that this miracle occurred?
It is a faith issue, definitely. No different than a belief in God. Who we say Mary is reveals to us who Christ is- fully God and fully man (hypostatic union). We call her Theotokos (God-bearer).
While I'm certain this will have no bearing on your belief on this issue, one way or the other, here is an interesting thing or two regarding what we Orthodox say about the Theotokos:
She's the ideal Christian- she is THE human we should look to so as to see what happens when we submit ourselves fully to Christ our God. She is a true handmaid of God. She attained theosis.
She is called the Unwedded Bride.
She gave birth to Christ- she is holy.
Her womb is more spacious than the heavens.
She is the ladder of divine descent.
She gave birth yet remained a virgin.
I'm totally borrowing this because it also explains how we see the Theotokos:
ORTHODIXIE ... Southern, Orthodox, Convert, Etc.: Mary, the Mother of God (Theotokos)
Appellations of the Theotokos.
Ark.
The Theotokos is often called an Ark, for the Glory of God settled on her, just as the Glory of God descended on the Mercy Seat of the Old Testament Ark of the Covenant (Ex. 25:10-22).
Aaron's Rod.
Just as Aaron's Rod sprouted miraculously in the Old Testament, so too, the Theotokos has budded forth the Flower of Immortality, Christ our God (Num. 17:1-11).
Burning Bush.
On Mt. Sinai, Moses saw the Bush that was burning, but was not consumed. So too, the Theotokos bore the fire of Divinity, but was not consumed (Ex. 3:1-6).
(Golden) Candlestick.
In the Old Testament Tabernacle, there were found in the Sanctuary golden candlesticks. The Theotokos is the Candlestick which held that Light that illumines the world (Ex. 25:31-40).
(Golden) Censer.
Just as the censer holds a burning coal, so too, the Theotokos held the Living Coal. In the Apocalypse, there stands an Angel before the Throne of God, swinging a censer, representing the prayers of the Saints rising up to God. This is also seen as a symbol of the Theotokos, for it is her prayers that find special favor before her Son.
Cloud.
In the Exodus, the Israelites were led out of Egypt by a Cloud of Light, symbolizing the presence of God in their midst. So too, the Theotokos is a Cloud, bearing God within.
Fleece.
In the book of Judges we read the account of the dew which appeared miraculously on Gideon's fleece (Judges 6:36-40). So too, the Dew Christ, appeared miraculously on the Living Fleece the Theotokos.
Holy of Holies.
Into the Holy of Holies only the High Priest could enter. So too, the Theotokos is the Holy of Holies into which only the Eternal High Priest Christ entered (Heb. 9:1-7).
Ladder.
In a dream Jacob saw a ladder ascending to Heaven, with Angels ascending and descending on it. The Theotokos is a Ladder, stretching from earth to Heaven, for on It God descended to man, having become incarnate.
Mountain (from which a Stone was cut not by hand of man).
The Prophet Daniel saw a mountain, from which was cut a stone, not by the hand of man (Dan. 2:34, 45). This is a reference to the miraculous Virgin Birth which was accomplished without the hand of man.
Palace.
The Theotokos was the Palace within which the King Christ our God dwelt.
Pot.
[See Urn]
Stem of Jesse.
In the Nativity Service, the Lord is referred to as the Rod from the Stem of Jesse (Is. 11:1), indicating His lineage from David, which was fulfilled through the Theotokos, who was a scion (or stem) of the line of David, the son of Jesse.
Tabernacle.
The Tabernacle was the place where the Glory of God dwelt. So too, the Glory of God dwelt in the Theotokos the Living Tabernacle (Ex. 40:34).
(Holy) Table.
This refers to the Holy Table (Altar Table) on which, at the Divine Liturgy, the Divine Food is offered. So too, the Theotokos is the Holy Table which bore the Bread of Life.
Temple.
The Prophet Ezekiel speaks of the Temple whose East gate remains sealed, through which only the Lord, the God of Israel, has entered. This clearly prophesies the Virgin Birth of the Theotokos (Ez. 44:1-2).
Throne.
The Theotokos is the Throne upon which Christ, the King of All, rested.
(Golden) Urn.
In the Old Testament, the Ark of the Covenant contained within itself a golden urn filled with the heavenly manna. The Theotokos is the Urn which contained Christ, the Divine Manna (Heb. 9:1-7).
Vine.
The Theotokos is the Vine which bore the Ripe Cluster (of Grapes), Christ our Lord.
