Why does Jesus give them the Holy Ghost?
Jesus breaths the Holy Ghost, as it were, into His Apostles, giving them the priestly ministry of forgiving sins by the same means He does: He says, "as the Father has sent Me, so also I am sending you," (in the same capacity, and with the same authority and mission) in reference to the ministry of forgiveness of sins, for which reasons He says, giving the reason they shall be able to forgive sins as Himself, "Recieve ye the Holy Ghost: whose sins you shall forgive, they shall be forgiven; whose sins you shall hold bound, they shall be kept bound."
This isn't a divinely condoned super-grudge. It is the priestly sacrament of confession (one can't discern between sins forgiven and ones which ought not to be without hearing those sins).
This helps us to understand the curious statement in the Gospel:
And entering into a boat, he passed over the water and came into his own city. 2 And behold they brought to him one sick of the palsy lying in a bed. And Jesus, seeing their faith, said to the man sick of the palsy: Be of good heart, son, thy sins are forgiven thee. 3 And behold some of the scribes said within themselves: He blasphemeth. 4 And Jesus seeing their thoughts, said: Why do you think evil in your hearts? 5 Whether is easier, to say, Thy sins are forgiven thee: or to say, Arise, and walk? 6 But that you may know that the Son of man hath power on earth to forgive sins, (then said he to the man sick of palsy,) Arise, take up thy bed, and go into thy house. 7 And he arose, and went into his house. 8 And the multitude seeing it, feared, and glorified God who gave such power to men.
The comment by Matthew is parenthetical, and his own, and is retrospective of the priestly mission given His Apostles, mere men—not anything the Jews said.
God, that is, Jesus, gave "such power to men," that is the power of granting absolution: "Son, thy sins are forgiven thee." Not by his own power or authority, but "by the Spirit of God" (Mt 12:28; Jn 20:22-23).
What does it mean He breatted the Holy Ghost upon them?
I suggest to you that Genesis 2:7 is referring to the Spirit of God by the term 'breath of life'), and not the breath, as we simply understand it, of life (interestingly, the word in both Hebrew and Greek for 'spirit' is that same of 'breath,' when an explicit word for 'breath' is not used—typologically, therefore, they are interchangeable).
For in Genesis 4:3 we read that "[God's] Spirit will not remain with[in] man forever: for he is mere flesh; and his days shall be [at most] a hundred and twenty years." That is, no doubt, the same Spirit He breathed into man at the beginning, "making him alive" (). (Hence the Nicaean-Constantinopolian credal statement, "the Holy Ghost, the Lord and Giver of life").
Undoubtedly, St. John is retaining this same imagery as is found in Genesis (as he does even in the opening verse of his Gospel) of God breathing His life-giving Breath into a creature, empowering him to 'live and have being' (Acts 17:28). His Gospel shows most clearly the divinity of Christ, and so this is probably just another one of the devices used by the writer to give nuance to the events of which he is making record, in order to call to mind precisely these things.
Speculation can buy us only more speculation, unfortunately, when it comes to imagining just how the scenes described in Sacred Scripture played out.