Note that this is an old ritual that only makes sense in the context in which it was developed. The ritual is no longer used, no doubt in part because it seems very strange for modern sensibilities.
A couple of clarifications are in order, first of all. Excommunication (and other ecclesiastical penalties) are, and always have been, "medicinal." Their purpose is to draw attention to a person who has committed a very grave crime (under ecclesiastical law), so that he may repent.
Excommunication specifically does not remove a person from membership in the Church (a common misconception), nor does it exclude one necessarily from eternal salvation (for example, in cases where the interested party repents, or else the penalty is applied in error or abusively). It does, however, exclude one from the reception of the Sacraments (although one may, of course, always receive the Sacrament of Reconciliation when in danger of death).
Canon 1331 §1 of the Code of Canon Law explains:
An excommunicated person is forbidden:
1° to have any ministerial participation in celebrating the sacrifice of the Eucharist or any other ceremonies of worship whatsoever;
2° to celebrate the sacraments or sacramentals and to receive the sacraments;
3° to exercise any ecclesiastical offices, ministries, or functions whatsoever or to place acts of governance.
Some of the ritual's more flamboyant expressions, such as the ones referred to by the original post, must, therefore, be considered as exaggerations to make a point.
I should point out that the ritual has numerous allusions to the Bible. For the benefit of readers, the full text of the rite is
Wherefore in the name of God the All-powerful, Father, Son, and Holy Ghost, of the Blessed Peter, Prince of the Apostles, and of all the saints, in virtue of the power which has been given us of binding and loosing in Heaven and on earth, we deprive him and all his accomplices and all his abettors of the Communion of the Body and Blood of Our Lord, we separate him from the society of all Christians, we exclude him from the bosom of our Holy Mother the Church in Heaven and on earth, we declare him excommunicated and anathematized and we judge him condemned to eternal fire with Satan and his angels and all the reprobate, so long as he will not burst the fetters of the demon, do penance and satisfy the Church; we deliver him to Satan to mortify his body, that his soul may be saved on the day of judgment.
"Binding and loosing" refers to Matthew 16:19
I will give you [Peter] the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven.”
or Matthew 18:18
Truly, I say to you, whatever you [apostles] bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven.
The "eternal fire with Satan and his angels" alludes to Matthew 25:41:
Then he will say to those on his left, "Depart from me, you cursed, into the eternal fire prepared for the devil and his angels."
Finally, the very last sentence is almost a direct quotation for 1 Cor 5:5:
You are to deliver this man to Satan for the destruction of the flesh, so that his spirit may be saved in the day of the Lord.
(Note how his quote emphasizes that, for all its flamboyancy, the purpose of the penalty is above to help the person receiving it to repent and return to communion with the Church.)
Another important point is that priests do not have, per se, the authority to excommunicate. Rather, the Church teaches, based on the passages from Matthew cited above, that only the competent authority in the Church has the authority to impose such a penalty. That means, practically, that only a diocesan bishop (or his equivalent: the "ordinary" of one of the Anglican-use ordinariates, for example). Naturally, whoever has authority over the entire Church may also impose such a penalty (for example, the Bishop of Rome—i.e., the Pope—or an ecumenical council, or possibly even a dicastery—a "congregation" at the Vatican—with delegated authority).
In any event, the Catholic Church does not claim to have authority over God, merely that God has bestowed upon the Church the power to "bind and loose"—to impose penalties on persons guilty of grievous crimes, and to re-admit them into communion if they repent.
So, analyzing the particular phrases in the original question:
... we exclude him from the bosom of our Holy Mother the Church in Heaven and on earth
This is to be understood as excluding the person above all from receiving the Sacraments, unless and until he repents. The ritual is, so to speak, assuming that the penalty is just and that the person is really guilty. The exclusion from communion is, as it were, only a public manifestation of what is going on internally in the person's soul.
we judge him condemned to eternal fire with Satan and his angels and all the reprobate,
Again, this presupposes that the person in question has really committed a grievous crime, and has done so out of malice. The idea is, if he persists in that state, then his eternal salvation will be lost.
we deliver him to Satan to mortify his body, that his soul may be saved on the day of judgment.
As I mentioned, this is almost a verbatim quotation from 1 Cor. 5:5, in which St. Paul prescribes a rather harsh treatment for a man who persists in a grave sexual sin:
It is actually reported that there is sexual immorality among you, and of a kind that is not tolerated even among pagans, for a man has his father's wife. And you are arrogant! Ought you not rather to mourn? Let him who has done this be removed from among you (1 Cor. 5:1-2).
In summary, the ritual described here uses the flamboyant language of the early Middle Ages. According to the custom of the time, since the ritual was pronounced after the person was put on trial, it presupposes the guilt of the person, and this comes out in some of its more extreme phrasings. Moreover, since excommunication is intended to help a person repent of a grave crime, language is used to help him understand the grave consequences of persisting in his depravity.