Swedenborg was a Christian
First and foremost, at the most basic level Emanuel Swedenborg (1688–1772) was a Christian.
As the son of an influential and iconoclastic Swedish Lutheran clergyman, Jesper Swedberg, Swedenborg grew up in an intensely Christian atmosphere. There is even some evidence in his personal journals that Swedenborg believed his father expected him, as the eldest surviving son, to follow in his father's footsteps and become a clergyman himself.
Swedenborg did have a very direct and personal relationship with Jesus Christ. This relationship became even more intense through a personal encounter with Jesus that he describes in the dream journal that he briefly kept in the years 1743 and 1744. These were pivotal years in his transition from being a scientist and philosopher to being a theologian. Many Swedenborg scholars believe that this encounter with Christ was the pivotal moment in that transition, completely changing the course of Swedenborg's life.
Based on his strongly Christian background and beliefs, and on his personal relationship with Jesus Christ, it would be unthinkable for Swedenborg not to view the Gospels, in which the life and teachings of Jesus Christ are recounted, as part of the Word of God—if not as the very heart of the Word of God.
And if there is any other book in the New Testament in which Jesus Christ appears most fully in person, it is the book of Revelation, with its opening vision of the risen and glorified Christ, and its (symbolic) account of Christ's battle against and ultimate victory over Satan.
By contrast, the Acts of the Apostles, as its traditional title suggests, focuses less on Jesus Christ himself and more on the activities of his immediate followers after his death.
And though the various Epistles, traditionally written by several of Christ's original Apostles plus the Apostle Paul, do teach many things about Jesus Christ, and amplify Christian doctrine in many ways, they don't have the immediacy of telling Christ's own story, and providing Jesus Christ's teachings in his own words.
On the heart level, then, Swedenborg held closely to the Gospels and the book of Revelation as the Word of God because these are the books in which his Lord, God, and Savior Jesus Christ is personally present, and speaks in his own words. This had a powerful meaning for Swedenborg, who had himself encountered Jesus Christ and heard Jesus' living voice speaking to him.
The books of the Word are those that have an internal sense
The more theological reason Swedenborg accepted the four Gospels and the book of Revelation as the Word of God he expressed in the quotation from Arcana Coelestia ("Secrets of Heaven") reproduced more fully my answer to the linked question:
The books of the Word are all those that have the internal sense; books that do not have it are not the Word. (Arcana Coelestia #10325)
Swedenborg lived in an era well before today's rampant biblical literalism had gotten a firm grip on much of the Christian Church. For him, it was a very basic part of Christian belief to think that the Bible, as the Word of God, had deeper meanings relating entirely to God and to our spiritual life.
In the years immediately following his call from Jesus Christ—which, based on Swedenborg's later references to it, occurred some time during the years 1743–1745—to set aside his previous scientific and philosophical labors and focus on the Bible and Christian theology instead, Swedenborg read and re-read the Bible in its original Hebrew and Greek languages, studying it intensively, indexing it thoroughly, and gaining an intimate, detailed knowledge of the book that had been held up to him as the Word of God from the time of his childhood.
In doing so, enlightened from within, he said, by the Lord himself, he began to see that not all of the books of the Protestant Bible had the same level of depth and inspiration.
More specifically, in the traditional Christian New Testament, he came to the conclusion that although the writers of and figures in the Acts and the Epistles did have a certain level of inspiration from the Holy Spirit, these books were written more from a historical and doctrinal perspective, and did not have the continuous, connected inner meaning that speaks at the deepest level entirely about the Lord's Incarnation and his process of glorification here on earth, and at a less deep level about our human rebirth or regeneration process from being self-centered sinners to being righteous people re-created by the Spirit of the Lord.
Another way of saying this is that the Acts and the Epistles explain the meaning of the life and teachings of Jesus Christ to their audience (Christians and potential Christians), whereas the Gospels and the book of Revelation tell the story of Jesus Christ.
Here is Swedenborg's own explanation of this difference in a letter that he wrote to one of his early followers:
As regards the writings of the Apostles and Paul, I have not quoted them in the Arcana Coelestia for the reason that they are doctrinal writings, and therefore are not written in the style of the Word as are the Prophets, David [the Psalms], the Gospels, and the Apocalypse [the book of Revelation]. The style of the Word consists wholly of correspondences, and therefore these books bring about direct communication with heaven.
In doctrinal writings there is a different style, which does indeed have communication with heaven, but indirectly. The reason these books were written in this way by the Apostles was so that through them the new Christian Church could have its beginning. To accomplish this purpose, doctrinal matters could not be written in the style of the Word, but rather had to be written in a way that could be understood more clearly and intimately.
The Apostles' writings are nevertheless good books for the church, maintaining the doctrine of charity and its faith just as strongly as does the Lord himself in the Gospels and the Apocalypse, as can be clearly seen and discovered if one has his mind on these matters when reading them. (Swedenborg's Letter to Beyer, April 15, 1766; translation edited into somewhat more contemporary English for readability)
By "correspondences" Swedenborg means the spiritual symbolism that he saw as existing throughout the inspired books of Scripture. The books of the inspired Word of God, Swedenborg said, are written in a style that contains this spiritual symbolism—or more accurately, this living relationship with spiritual and divine things—from beginning to end.
The Acts and the Epistles in the New Testament, by contrast, he saw as having a more historical and doctrinal style that, although it did contain some spiritual symbolism, was not fully "correspondential" or symbolic (what today is often called "metaphorical") specifically so that it could explain Christian doctrine more clearly and directly to the members of the nascent Christian Church.
And so, as stated in my linked answer, Swedenborg did not reject the Acts and the Epistles. Indeed, he believed that they had a critical role in the founding and building up of the Christian Church. He simply didn't see them as providing, at a deeper level, a continuous and coherent narrative, from beginning to end, of Jesus Christ's spiritual and divine work during his lifetime on earth, and of our own spiritual rebirth process.
For a full list of the books that Swedenborg saw as the inspired Word of God in both the Old Testament and the New Testament, see my answer to the linked question.