Fr John Flader: How did Jesus actually rise again?

Q: In some places in the New Testament it says that it was the Father
who raised Jesus from the dead. I thought he rose by his own power. Can
you please explain?

You are right in saying that there are passages in the New Testament that say that Jesus was raised by the Father.
For example, St Peter says to the multitudes gathered in Jerusalem on the day of Pentecost: “But God raised him up, having loosed the pangs of death, because it was not possible for him to be held by it” (Acts 2:24).
And St Paul writes to the Romans: “We were buried therefore with him by Baptism into death, so that as Christ was raised from the dead by the glory of the Father, we too might walk in newness of life” (Rom 6:4).
From passages like these someone might be led to believe that the Resurrection of Jesus was not really remarkable and that Jesus was only a man after all. They could think that just as Jesus himself raised three people from the dead, so the Father raised Jesus from the dead.
Before going further, we should remember that Jesus’ Resurrection, independently of how it came about, was very different from that of the three people Jesus raised from the dead.
When Jesus raised Lazarus (cf. Jn 11:38-44), the son of the widow of Naim (cf. Lk 7:11-17) and the daughter of Jairus, the ruler of the synagogue (cf. Lk 8:40-42, 49-56), each of these people returned to their natural state, as if they had just woken up from a deep sleep. They would continue their life with their loved ones and eventually would have to die again.
Jesus’ Resurrection was different. He did not rise in the natural state he had when he died on the Cross.
Before his death, he was subject to space and time, and was always present somewhere on earth. After his Resurrection, to be sure, he rose in the same body he had before, so that he could show the apostles in the Upper Room the wounds in his hands and his feet (cf. Lk 24:39) and he could invite Thomas to put his finger into the wounds (cf. Jn 20:24-27).
But his body was not subject to space and time in the way it was before. Now he could suddenly appear and disappear, as he did with the disciples on their way to Emmaus (cf. Lk 24:13-31), or when he passed through locked doors and appeared to the apostles in the Upper Room (cf. Lk 24:36, Jn 20:19).
Also, he was sometimes not recognised, even though he was well known to people like Mary Magdalene (cf. Jn 20:14-15) and the disciples of Emmaus (cf. Lk 24:16).
In the words of the Catechism of the Catholic Church, “…this authentic, real body possesses the new properties of a glorious body: not limited by space and time but able to be present how and when he wills.” (CCC 645)
Returning to your question, even though there are passages that speak of Jesus being raised by the Father, it is clear that he rose by his own power. The Roman Catechism, issued after the Council of Trent, says: “By the word ‘resurrection’ we are not merely to understand that Christ was raised from the dead … but that he rose by his own power and virtue, a singular prerogative peculiar to him alone.
Our Lord confirmed this by the divine testimony of his own mouth when he said: ‘I lay down my life, that I may take it again … I have power to lay it down: and I have power to take it up again’ (Jn 10:17-18).
“To the Jews he also said, in corroboration of his doctrine: ‘Destroy this temple, and in three days I will raise it up’ (Jn 2:19-20)… We sometimes, it is true, read in Scripture that he was raised by the Father (cf. Acts 2:24: Rom 8:11); but this refers to him as man, just as those passages on the other hand, which say that he rose by his own power, relate to him as God” (St Pius V, Catechism, 1, 6, 8).
The fact that Jesus rose by his own power, after prophesying that he would rise on the third day (cf. Lk 18:33), is the ultimate proof of his divinity. Only God can raise himself from the dead.
The Catechism sums it up: “The Resurrection of Jesus is the crowning truth of our faith in Christ, a faith believed and lived as the central truth by the first Christian community; handed on as fundamental by Tradition; established by the documents of the New Testament…” (CCC 638)