If entering the Catholic church is not possible after death, what purpose do prayers for the dead serve?
There have been a few misconceptions in this thread, not the least of which is that the priest who was himself involved in this accident, was an Anglican priest not a Catholic. He was also humbly offering a prayer of blessing for those who were injured (mortally or not), not necessarily performing Last Rites (or Anointing of the Sick).
OT
2 Maccabees 12:42-46. 41 Then they all blessed the just judgment of the Lord, who had discovered the things that were hidden. 42 And so betaking themselves to prayers, they besought him, that the sin which had been committed might be forgotten. But the most valiant Judas exhorted the people to keep themselves from sin, forasmuch as they saw before their eyes what had happened, because of the sins of those that were slain. 43 And making a gathering, he sent twelve thousand drachms of silver to Jerusalem for sacrifice to be offered for the sins of the dead, thinking well and religiously concerning the resurrection, 44 (For if he had not hoped that they that were slain should rise again, it would have seemed superfluous and vain to pray for the dead,) 45 And because he considered that they who had fallen asleep with godliness, had great grace laid up for them.
Ecclesiasticus 7:37. A gift hath grace in the sight of all the living, and restrain not grace from the dead.
Zacharias 9:11. 11 Thou also by the blood of thy testament hast sent forth thy prisoners out of the pit, wherein is no water.
Micheas 7:8-9. 8 Rejoice not, thou, my enemy, over me, because I am fallen: I shall arise, when I sit in darkness, the Lord is my light. 9 I will bear the wrath of the Lord, because I have sinned against him; until he judge my cause and execute judgment for me: he will bring me forth into the light, I shall behold his justice.
...and was continued by the Christians in the New Testament
NT
Matthew 5:25-26. 25 Be at agreement with thy adversary betimes, whilst thou art in the way with him: lest perhaps the adversary deliver thee to the judge, and the judge deliver thee to the officer, and thou be cast into prison. 26 Amen I say to thee, thou shalt not go out from thence till thou repay the last farthing.
Matthew 12:32,36. 32 And whosoever shall speak a word against the Son of man, it shall be forgiven him: but he that shall speak against the Holy Ghost, it shall not be forgiven him, neither in this world, nor in the world to come. 36 But I say unto you, that every idle word that men shall speak, they shall render an account for it in the day of judgment.
1 Peter 3:18-19. 18 Because Christ also died once for our sins, the just for the unjust: that he might offer us to God, being put to death indeed in the flesh, but enlivened in the spirit, 19 In which also coming he preached to those spirits that were in prison
Apocalypse 21:27. 27 There shall not enter into it any thing defiled
The above passages describe the belief of purgatory, a state of cleansing which one must undergo to enter into Heaven. Some endure purgatory on earth and go straight to Heaven when they die. Receiving the Sacraments before death can often mean the difference between Hell and Purgatory or Purgatory and Heaven for a Catholic. Saints in Heaven don't need our prayers and the souls in Hell cannot benefit. Therefore the prayers for the dead are for the souls in purgatory to help shorten their time of cleansing.
To answer your question directly, when Catholics "pray for the dead", they/we are praying for those who's bodies have ceased to function, but yet for reasons of hope, are believed to be "alive in Christ" insomuch as they didn't die with any mortal sin against them.
The "prayers for the dead" are to help them in the purification process of Purgatory. Some believe Purgatory to be painful as physical pain is for us now, others, believe it's a spiritual pain (pain of still not yet being in the full presence of the Lord) but either way, whichever really occurs the prayers of the faithful on Earth help ease the suffering of the souls in Purgatory. So that's what the prayers for the dead said at every Mass (and any other time) are for: to alleviate the suffering of their soul in Purgatory and/or to shorten their time there.
I'm not really sure there are Catholic priests that would have done what this Anglican did, as our "prayers for the dead" are not the same thing as the Sacrament of Last Rites, but I could be wrong (IOW, I'm not sure if Catholic priests would attempt to give Last Rites to a person who appeared to be dead). Again, I could be wrong. I'm just saying that though to make a clear distinction between the two concepts.