Two different Greek Church Fathers - Gregory of Nyssa (335-394) and Gregory of Nazianzus (320-390) - offer two slightly different but not incompatible interpretations of what is meant by firstborn among many brethren (πρωτοτοκον εν πολλοις αδελφοις).
Gregory of Nyssa explains πρωτοτοκον εν πολλοις αδελφοις as referring to Christ's Baptism:
Assuredly this is plain, that because we are by birth flesh and blood,
as the Scripture saith, He Who for our sakes was born among us and
was partaker of flesh and blood [Hebrews 1:14], purposing to change us from
corruption to incorruption by the birth from above, the birth by water
and the Spirit, Himself led the way in this birth, drawing down upon
the water, by His own baptism, the Holy Spirit; so that in all things
He became the first-born of those who are spiritually born again, and
gave the name of brethren to those who partook in a birth like to His
own by water and the Spirit.
Against Eunomius, Book II, No. 8
Gregory of Nazianzus explains the phrase as relating to three different births: the natural birth, Baptism, and the Resurrection.
The Word recognizes three Births for us; namely, the natural birth,
that of Baptism, and that of the Resurrection. Of these the first is
by night, and is servile, and involves passion; but the second is by
day, and is destructive of passion, cutting off all the veil [i.e.
ancestral sin] that is derived from birth, and leading on to the
higher life; and the third is more terrible and shorter, bringing
together in a moment all mankind, to stand before its Creator, and to
give an account of its service and conversation here; whether it has
followed the flesh, or whether it has mounted up with the spirit, and
worshipped the grace of its new creation. My Lord Jesus Christ has
showed that He honoured all these births in His own Person; the first,
by that first and quickening Inbreathing; [Genesis 2:7] the second by
His Incarnation and the Baptism wherewith He Himself was baptized; and
the third by the Resurrection of which He was the Firstfruits;
condescending, as He became the Firstborn [Romans 8:29] among many
brethren, so also to become the Firstborn from the dead. [Colossians
Oration XL, "On Holy Baptism"
Note that Gregory of Nyssa here is referring to the Holy Spirit coming down upon the water - not upon Christ. It has been suggested that Christ received the Holy Spirit at Baptism, but this would be considered heresy by Nicene-Constantinopolitan Christians (i.e. those who accept the Creed of Nicene-Constantinople), since it would imply that Christ's Divinity was somehow incomplete at some point while He was Incarnate. The purpose of Christ's Baptism, according to the Church Fathers at least, was to make the Trinity manifest and not to somehow complete it. Christ's Baptism sanctified the water, it was not the water that sanctified Christ.