From a Swedenborgian perspective, there is a simple answer and a complex answer to this question.

The Simple Answer

The simple answer does not require the theology of Emanuel Swedenborg (1688-1772) at all. It only requires common human experience:

Jesus was talking to himself.

People often talk to themselves. Sometimes they even do it out loud! I doubt anyone goes through life without talking to himself or herself on many occasions—perhaps many times a day. The Bible records a number of times when people talk to themselves. A simple search for "said to himself" on BibleGateway turns up seven instances in the Protestant Bible, and three more in the Apocrypha.

But more specifically from a Swedenborgian perspective:

Jesus was talking to his own soul.

In Swedenborg's theology, "the Father" is the divine soul, while "the Son" is the divine body, or human manifestation. (See my answer to the question, "What do non-trinitarians mean when they call Jesus the 'Son of God'?")

The Bible also offers an example of someone talking to his own soul, in Jesus' parable of the rich fool:

Then he told them a parable: "The land of a rich man produced abundantly. And he thought to himself, 'What should I do, for I have no place to store my crops?'

"Then he said, 'I will do this: I will pull down my barns and build larger ones, and there I will store all my grain and my goods. And I will say to my soul, Soul, you have ample goods laid up for many years; relax, eat, drink, be merry.'" (Luke 12:16-19, italics added)

So the simple answer, which doesn't even require Swedenborg's theology, but only a common understanding of human nature (which is an image and likeness of God's nature, see Genesis 1:26-27), is:

Jesus was talking to himself, and specifically, to his own soul, which was the Father within him.

The Complex Answer

The complex answer does require Swedenborg's theology of the Incarnation, which also has a solid basis in the Bible.

Jesus' dual nature while on earth

According to the two Biblical accounts of Jesus' conception and birth, which occur in Matthew 1:18-25 and Luke 1:26-38; 2:1-7, Mary was Jesus' mother. However his father was not Joseph. Instead Mary conceived him "from the Holy Spirit," such that the child to be born of her would be called "the Son of God." The clear inference is that God was Jesus' father. And the New Testament commonly refers to God as the Father of Jesus.

This means that at birth, Jesus had a dual nature:

1. A finite human nature from his human mother Mary
2. An infinite divine nature from his divine Father

These two natures were arranged in him such that the finite human nature from Mary constituted his outer self, while his infinite divine nature, which was God, constituted his inner self. Swedenborg states this succinctly in Secrets of Heaven #1460:

His inner aspects were divine ones from Jehovah, his Father; his outer aspects were human ones from Mary, his mother.

According to Swedenborg's theology, it was through his finite human nature, which he received from Mary, that Jesus could be tempted by the Devil.

In fact, it was the very weaknesses and tendencies toward sin (not Original Sin—a non-Biblical Catholic doctrine that Swedenborg rejected) that made it possible for hell and the Devil to approach Jesus so that he could be "tempted in all ways just as we are" (Hebrews 4:15). The Devil could never approach God's divine nature directly. Such an encounter would annihilate the Devil—who, in Swedenborg's theology is a personification of all evil and hell.

Ironically then, according to Swedenborg's theology, the non-Biblical Catholic doctrine of Mary's Immaculate Conception (which is not the same thing as the Virgin Birth of Jesus), if it were true, would destroy the very reason Jesus took on a human nature through and from a finite human being. He took on that nature specifically to face the Devil on the Devil's own turf, which is the fallen human nature, subject to sin. In other words, the Incarnation required that Jesus receive from his human mother a finite, fallen human nature with inborn tendencies to evil and sin just like that of any other human being ever born.

Jesus' victory over the Devil through temptation

And yet, according to Hebrews 4:15, though he was tempted to sin in every way that we are, unlike any ordinary human being he never actually sinned. In line with this, Swedenborg states that during Jesus' lifetime on earth, every time the Devil approached him through his finite, human nature and tempted him, Jesus overcame in that temptation, and was victorious in his struggle against the Devil (or hell). This can be seen especially in Jesus' temptation by the Devil in the desert after his baptism (Matthew 4:1-11; Mark 1:12-13; Luke 4:1-13).

Through a lifetime of victories in all of his temptations, Jesus completely overcame the Devil, or the power of evil, subjecting it to his personal control, and reducing it to a state in which it could no longer overwhelm humanity.

This, according to Swedenborg was the essential and universal act of Redemption that Jesus accomplished during his lifetime on earth. If Jesus had not fought against and emerged completely victorious over the Devil, all people on earth would have suffered eternal damnation, because by our own strength we humans cannot possibly resist the overwhelming power of the Devil and hell.

Jesus' process of glorification

This victory over evil took place in the context of a process that Swedenborg calls "glorification," based on the usage of that word especially in the Gospel of John. (See, for example, John 7:39; 12:23, 28; John 13:31-32; 14:13.)

This process of glorification took place in two basic states that continued throughout his lifetime in an alternating cycle:

1. The first state was one of emptying out, or of expelling from himself the finite human nature from his human mother Mary.
2. The second state was one of being glorified, or of uniting his human nature with his inner divine nature, called "the Father" in the Bible.

Here is how Swedenborg describes this process, and its ultimate results, in True Christianity #104:

When the Lord was being emptied out he was in a state of progress toward union; when he was being glorified he was in a state of union itself. The church recognizes that the Lord had two states while he was in the world: one called being emptied out; the other called glorification.

The prior state, being emptied out, is described in many passages in the Word, especially in the Psalms of David, but also in the Prophets. There is even one passage in Isaiah 53 where it says, "He emptied out his soul even to death" (Isaiah 53:12). This same state also entailed the Lord's being humbled before the Father. In this state he prayed to the Father. In this state he says that he is doing the Father’s will and attributes everything he has done and said to the Father.

The following passages show that he prayed to the Father: Matthew 26:36-44; Mark 1:35; 6:46; 14:32–39; Luke 5:16; 6:12; 22:41-44; John 17:9, 15, 20. The following show that he did the Father's will: John 4:34; 5:30. The following show that he attributed everything he had done and said to the Father: John 8:26, 27, 28; 12:49, 50; 14:10.

In fact, he cried out on the cross, "My God, my God, why have you abandoned me?" (Matthew 27:46; Mark 15:34). Furthermore, without this state it would have been impossible to crucify him.

The state of being glorified is also a state of union. The Lord was in this state when he was transfigured before three of his disciples. He was in it when he performed miracles. He was in it as often as he said that the Father and he were one, that the Father was in him and he was in the Father, and that all things belonging to the Father were his. After complete union he said he had power over all flesh (John 17:2) and all power in heaven and on earth (Matthew 28:18). There are also other such passages. (links added)

Why Jesus sometimes talked and prayed to the Father as if to a separate being

In other words, during his lifetime on earth Jesus went through two alternating states:

• In one state, Jesus was more present in and aware of his finite human nature from Mary. This was the state in which he could be tempted. It was also a state in which he felt a sense of separation from the Father, who was his own inner divine soul. It was in this state that Jesus prayed to the Father as if to a separate being.
• In the other state, Jesus was more present in and aware of his infinite divine nature, which was God the Father. When he was in this state, he said that he and the Father are one (John 10:30), that those who have seen him have seen the Father (John 14:8-9), and that the Father within him does the works (John 14:10).

These alternating states of feeling separated from the Father (his own divine soul) when he was more conscious of his finite human side, and feeling one with the Father when he was more conscious of his infinite divine side, continued throughout his lifetime on earth, culminating in his final and greatest temptation, on the Cross, when he cried out (quoting the opening verse of Psalm 22), "My God, my God, why have you forsaken me?" (Matthew 27:46; Mark 15:34).

Jesus' full union with the Father

That, according to Swedenborg, was the last time he was immersed in his finite human nature from Mary. Through his victory in that final and greatest temptation, he completely put off from himself the last vestiges of his finite human heredity from Mary, and fully united himself with the divinity of the Father within. After the Resurrection, and especially after the Ascension to the Father, there was nothing left of that finite human maternal heredity; he was fully divine, so that Swedenborg calls him the Lord God Jesus Christ.

One final passage illustrating this, from Swedenborg's True Christianity #102:

There is a belief that the Lord in his human manifestation not only was but still is the Son of Mary. This is a blunder, though, on the part of the Christian world. It is true that he was the Son of Mary; it is not true that he still is. As the Lord carried out the acts of redemption, he put off the human nature from his mother and put on a human nature from his Father. This is how it came about that the Lord's human nature is divine and that in him God is human and a human is God. The fact that he put off the human nature from his mother and put on a divine nature from his father—a divine human nature—can be seen from his never referring to Mary as his mother, as the following passages show: "The mother of Jesus said to him, 'They have no wine.' Jesus said to her, 'What do I have to do with you, woman? My hour has not yet come'" (John 2:3-4). Elsewhere it says, "Jesus on the cross saw his mother and the disciple he loved standing next to her. He said to his mother, 'Woman, behold your son.' Then he said to the disciple, 'Behold your mother'" (John 19:26-27). On one occasion he did not acknowledge her: "There was a message for Jesus from people who said, 'Your mother and your brothers are standing outside, and they want to see you.' Jesus said in reply, 'My mother and my brothers are these people who are hearing the Word of God and doing it'" (Luke 8:20-21; Matthew 12:46-49; Mark 3:31-35). So the Lord called her "woman," not "mother," and gave her to John to be his mother. In other passages she is called his mother, but not by the Lord himself.

So although Jesus did have a dual nature during his lifetime on earth, having a finite human side outwardly, but an infinite divine side inwardly, after his process of glorification on earth was complete, he was fully divine and human at the same time, his humanity having become fully divine. This is why another common expression that Swedenborg uses to describe the risen and glorified Jesus Christ is "the Divine Humanity."

So the complex answer is:

When Jesus talked and prayed to the Father as if to a separate being, he was doing so from the finite, human nature that he received from his mother Mary. However, by the time of his Resurrection and Ascension, none of that that finite nature was left in him. He had become fully divine, and fully one with the Father, so that Father, Son, and Holy Spirit are all essential components of one Divine Person, who is the Divine Humanity, or the Lord God and Savior Jesus Christ.