To see how Jesus might've escaped Adam's original sin, we must first understand what "original sin" means in the first place.
The original sin refers to the sin Adam and Eve committed when they disobeyed God and ate the forbidden fruit, therefore passing their sin, or rather their tendency to sin, to all of their descendants—all humans. This original sin explains the existence of all evil in the world.
Some believe that Adam and Eve were the ones who brought evil to the world when they disobeyed God, while others believe that evil was already there as an opposing force to good, and Adam and Eve introduced evil and the choice to disobey God to humanity by falling into temptation. The premise remains, however, that all humans are born sinful and allow evil to exist in their world as a consequence to Adam and Eve's original sin.
If all descendants of Adam and Eve are truly born into sin or with the urge to sin, then how did Jesus manage to escape that sin?
The answer to that question might be connected to Jesus's nature, and suggests that Jesus managed to escape original sin because he was not only human, but also God; he had a human mother, but no biological human father and his birth was a miracle on its own, as he was conceived with the help of the Holy Spirit.
Jesus had all the physical and mental attributes of a man, but his soul was divine. As a man, he was tempted to sin and perhaps even had the urge to disobey God, but as someone who had the power, will, and grace of God in him, he was capable of doing what Adam, Eve, and all humans were incapable of doing—he resisted temptation. Jesus might've been as "weak" as all humans when it came to temptation and desire, but he also had the divine power to overcome them, which no other individual possessed:
For we do not have a high priest who is unable to empathize with our weaknesses, but we have one who has been tempted in every way, just as we are—yet he did not sin. (Hebrews 4:15)
In the end, Jesus died not as a punishment for his sins, but for the sins of humanity; if Adam and Eve introduced sin to humans, Jesus managed to redeem them from that sin.