Why Don't We Remember Our Past Lives?

There is no doubt that Hashem gives people who seek Him the opportunity to find Him and return in repentance. And one who returns in repentance surely learns that they must especially rectify what is difficult for them to fix, as the Gaon of Vilna said, everyone can know precisely what they descended into the world to amend. One should see what mitzvot are hardest for them to observe and realize that's the area they've returned to rectify. The parts of the Torah and mitzvot they are drawn to study and observe heal their soul the most.

Nevertheless, it's also certain that knowledge of past incarnations would greatly ease our struggles with the suffering in this world, because then we would know exactly why we hurt, and how much we have left to fix, like a person in prison who knows how many years they must serve. There's no doubt it's easier for a prisoner to serve their sentence knowing their exact crime, than for one who feels innocent and doesn't know their offense. I haven't heard a definitive answer to this question, so I suggest several answers that have come to mind:

1. One answer is to lessen the punishment:

There is no doubt that divine calculations are hidden from us in this world. However, in my humble opinion, the reason a person doesn't know their past life is to lessen the intensity of their punishment. Logic dictates that someone who knows their sin must be punished more than someone who doesn't, because ignorance is itself at least half the punishment! So Hashem brings the person back to this world without awareness, so they aren't forced to face multiple times the punishment in Gehenna.

2. A second answer that comes to mind is punishment measure for measure:

Just as a person caused harm to another without cause, so too that person will feel in this world (after reincarnation) that they are punished without cause. They are punished exactly according to the severity of their harm—emotionally as well, measure for measure—and thus their atonement is complete and absolute.

3. To increase a person’s reward and their standing in the World to Come: I believe this is also a test of faith, how much a person trusts in Hashem who desires their good. For this test, a person receives eternal reward in Gan Eden. This is also a purpose of returning in reincarnation—to increase one's reward in the World to Come. For life here is a test. Instead of being punished in the World to Come and remaining in their status, one returns to the world to rectify sins and simultaneously increase their reward in the World to Come through the power of faith.

4. To merit compassion: I once heard that Hashem hides a person’s actions from past lives so they will merit compassion from others. If people saw a certain beggar on the street and knew that in a past life they sinned in theft or something similar, or saw a sick person knowing they greatly harmed another in a past life, they wouldn't feel compassion and help as much, and the world would lose its merciful nature towards the afflicted and vulnerable. This was also the sin of Job's friends, because they lived in a time of open divine providence and were completely sure Job sinned grievously, hidden from them, they hurt his feelings until he told them, his suffering isn't equivalent to the accusations they made! A person's honor is sometimes more important than the pain they endure, and one is willing to suffer much for their dignity. Thus, Hashem hid the secret of reincarnations so we would all be equal in this world in our ignorance of our pasts, and would have compassion on another, give charity, and feel for each other as we feel for ourselves, fulfilling "love your neighbor as yourself."

5. The well-known answer agreed upon by all is to ensure full free choice in this world: because if a person knew clearly what their actions were in a past incarnation, they wouldn't sin, and out of fear their free choice would be nullified. For what is faith, and what is the challenge, if a person knows everything with absolute certainty?

I recommend reading the booklet "Critical Conversation" (search on Google), which discusses at length the topic of suffering and the purpose of creation.