What are the traditional reasons ascribed to why God saw it fit that Jesus be born of a virgin? According to the early church fathers, why did God decide that Jesus would be born of a virgin? I would imagine that someone like Augustine would say that it has something to do with why he is unstained from original sin, but the Bible does not explicitly say this. What are some of the early interpretations?
In the Catholic Encyclopedia you can find a thorough historical and theological analysis (up until 1912 at least) of the development of the virgin birth doctrine/belief, including a list of early church fathers that wrote on the topic. The list is (for links to authors/text, see website):

St. Irenaeus (III, 21; see Eusebius, Church History V.8),
Origen (Adv. Cels., I, 35),
Tertullian (Adv. Marcion., III, 13; Adv. Judæos, IX),
St. Justin (Dialogue with Trypho 84),
St. John Chrysostom (Hom. v in Matth., n. 3; in Isa., VII, n. 5);
St. Epiphanius (Hær., xxviii, n. 7),
Eusebius (Demonstrat. ev., VIII, i),
Rufinus (Lib. fid., 43),
St. Basil (in Isa., vii, 14; Hom. in S. Generat. Christi, n. 4, if St. Basil be the author of these two passages),
St. Jerome and Theodoretus (in Isa., vii, 14),
St. Isidore (Adv. Judæos, I, x, n. 3),
St. Ildefonsus (De perpetua virginit. s. Mariæ, iii).
St. Jerome devotes his entire treatise against Helvidius to the perpetual virginity of Our Blessed Lady (see especially nos. 4, 13, 18).

In effect, several sects and individuals arose in the first centuries denying the virginal birth of Jesus. For example, (some of the) Ebionites and Cerinthus. Against them wrote the early father St. Irenaeus of Lyons in his book Against heresies (the first on in the list above). In Chapter 21 of Book III, for example, he states (emphasis mine):


But what Isaiah said, &quot;From the height above, or from the depth beneath,&quot; Isaiah 7:11 was meant to indicate, that &quot;He who descended was the same also who ascended.&quot; Ephesians 4:10 But in this that he said, &quot;The Lord Himself shall give you a sign,&quot; he declared an unlooked-for thing with regard to His generation, which could have been accomplished in no other way than by God the Lord of all, God Himself giving a sign in the house of David. For what great thing or what sign should have been in this, that a young woman conceiving by a man should bring forth — a thing which happens to all women that produce offspring? But since an unlooked-for salvation was to be provided for men through the help of God, so also was the unlooked-for birth from a virgin accomplished; God giving this sign, but man not working it out.


Later on, he says (emphasis mine):

And as the protoplast himself Adam, had his substance from untilled and as yet virgin soil (&quot;for God had not yet sent rain, and man had not tilled the ground&quot; Genesis 2:5), and was formed by the hand of God, that is, by the Word of God, for &quot;all things were made by Him,&quot; John 1:3 and the Lord took dust from the earth and formed man; so did He who is the Word, recapitulating Adam in Himself, rightly receive a birth, enabling Him to gather up Adam [into Himself], from Mary, who was as yet a virgin. If, then, the first Adam had a man for his father, and was born of human seed, it were reasonable to say that the second Adam was begotten of Joseph. But if the former was taken from the dust, and God was his Maker, it was incumbent that the latter also, making a recapitulation in Himself, should be formed as man by God, to have an analogy with the former as respects His origin. Why, then, did not God again take dust, but wrought so that the formation should be made of Mary? It was that there might not be another formation called into being, nor any other which should [require to] be saved, but that the very same formation should be summed up [in Christ as had existed in Adam], the analogy having been preserved.


Although not an Early Church father, St. Thomas Aquinas deals with this very directly in his Summa Theologicae, quoting Agustine in a few paragraphs:

It is fitting for four reasons that Christ should be born of a virgin. First, in order to maintain the dignity or the Father Who sent Him. For since Christ is the true and natural Son of God, it was not fitting that He should have another father than God: lest the dignity belonging to God be transferred to another.
Secondly, this was befitting to a property of the Son Himself, Who is sent. For He is the Word of God: and the word is conceived without any interior corruption: indeed, interior corruption is incompatible with perfect conception of the word. Since therefore flesh was so assumed by the Word of God, as to be the flesh of the Word of God, it was fitting that it also should be conceived without corruption of the mother.
Thirdly, this was befitting to the dignity of Christ's humanity in which there could be no sin, since by it the sin of the world was taken away, according to John 1:29: &quot;Behold the Lamb of God&quot; (i.e. the Lamb without stain) &quot;who taketh away the sin of the world.&quot; Now it was not possible in a nature already corrupt, for flesh to be born from sexual intercourse without incurring the infection of original sin. Whence Augustine says (De Nup. et Concup. i): &quot;In that union,&quot; viz. the marriage of Mary and Joseph, &quot;the nuptial intercourse alone was lacking: because in sinful flesh this could not be without fleshly concupiscence which arises from sin, and without which He wished to be conceived, Who was to be without sin.&quot;
Fourthly, on account of the very end of Incarnation of Christ, which was that men might be born again as sons of God, &quot;not of the will of the flesh, nor of the will of man, but of God&quot; (John 1:13), i.e. of the power of God, of which fact the very conception of Christ was to appear as an exemplar. Whence Augustine says (De Sanct. Virg.): &quot;It behooved that our Head, by a notable miracle, should be born, after the flesh, of a virgin, that He might thereby signify that His members would be born, after the Spirit, of a virgin Church.&quot;