Why is a soul reincarnated, and the purpose of reincarnation?

There are some suttas providing info on this like MN 135 (quoted below), MN 136 and AN 4.85.

However, as mentioned in other answers, there is rebirth in the suttas, but no reincarnation. And certainly there is no soul or impermanent self. According to MN 38, there is no continuity of the same consciousness throughout one or more lives.

Please see this answer for an explanation.

Rebirth is only the continuation of chained conditioned processes and the continuation of suffering. We don't assign any identity or self or soul to this rebirth.

However, the rebirth-of-self view is also used as skillful means or as a tool to get unenlightened persons with self view, to instill the mindset of being ashamed of doing evil (hiri) and having a fear of doing evil (otappa). This is found in AN 5.57 and in line with the Right View with effluents or taints in MN 117.

The Blessed One said: "There is the case, student, where a woman or man is a killer of living beings, brutal, bloody-handed, given to killing & slaying, showing no mercy to living beings. Through having adopted & carried out such actions, on the break-up of the body, after death, he/she reappears in the plane of deprivation, the bad destination, the lower realms, hell. If, on the break-up of the body, after death — instead of reappearing in the plane of deprivation, the bad destination, the lower realms, hell — he/she comes to the human state, then he/she is short-lived wherever reborn. This is the way leading to a short life: to be a killer of living beings, brutal, bloody-handed, given to killing & slaying, showing no mercy to living beings.

"But then there is the case where a woman or man, having abandoned the killing of living beings, abstains from killing living beings, and dwells with the rod laid down, the knife laid down, scrupulous, merciful, & sympathetic for the welfare of all living beings. Through having adopted & carried out such actions, on the break-up of the body, after death, he/she reappears in a good destination, in the heavenly world. If, on the break-up of the body, after death — instead of reappearing in a good destination, in the heavenly world — he/she comes to the human state, then he/she is long-lived wherever reborn. This is the way leading to a long life: to have abandoned the killing of living beings, to abstain from killing living beings, to dwell with one's rod laid down, one's knife laid down, scrupulous, merciful, & sympathetic for the welfare of all living beings.

"There is the case where a woman or man is one who harms beings with his/her fists, with clods, with sticks, or with knives. Through having adopted & carried out such actions, on the break-up of the body, after death, he/she reappears in the plane of deprivation... If instead he/she comes to the human state, then he/she is sickly wherever reborn. This is the way leading to sickliness: to be one who harms beings with one's fists, with clods, with sticks, or with knives.

"But then there is the case where a woman or man is not one who harms beings with his/her fists, with clods, with sticks, or with knives. Through having adopted & carried out such actions, on the break-up of the body, after death, he/she reappears in a good destination... If instead he/she comes to the human state, then he/she is healthy wherever reborn. This is the way leading to health: not to be one who harms beings with one's fists, with clods, with sticks, or with knives.

"There is the case, where a woman or man is ill-tempered & easily upset; even when lightly criticized, he/she grows offended, provoked, malicious, & resentful; shows annoyance, aversion, & bitterness. Through having adopted & carried out such actions, on the break-up of the body, after death, he/she reappears in the plane of deprivation... If instead he/she comes to the human state, then he/she is ugly wherever reborn. This is the way leading to ugliness: to be ill-tempered & easily upset; even when lightly criticized, to grow offended, provoked, malicious, & resentful; to show annoyance, aversion, & bitterness.

"But then there is the case where a woman or man is not ill-tempered or easily upset; even when heavily criticized, he/she doesn't grow offended, provoked, malicious, or resentful; doesn't show annoyance, aversion, or bitterness. Through having adopted & carried out such actions, on the break-up of the body, after death, he/she reappears in a good destination... If instead he/she comes to the human state, then he/she is beautiful wherever reborn. This is the way leading to beauty: not to be ill-tempered or easily upset; even when heavily criticized, not to be offended, provoked, malicious, or resentful; nor to show annoyance, aversion, & bitterness.

"There is the case where a woman or man is envious. He/she envies, begrudges, & broods about others' gains, honor, respect, reverence, salutations, & veneration. Through having adopted & carried out such actions, on the break-up of the body, after death, he/she reappears in the plane of deprivation... If instead he/she comes to the human state, then he/she is not influential wherever reborn. This is the way leading to not being influential: to be envious, to envy, begrudge, & brood about others' gains, honor, respect, reverence, salutations, & veneration.
MN 135

The difference with the Buddhist notion of reincarnation (lit. "re-enfleshment") is that the two incarnations do not share the same mind, nor does the same mind physically travel between physical bodies. There is no notion of "the soul" in Buddhism. Instead, there is "the mind," "the body," and "the life faculty."

"The soul" is a Judeo-Christian concept. Certainly, other cultures have things in them that resemble "the soul" (such as the "po and hum" of traditional Chinese religion, or the "Ka" of ancient Egyptian religion, etc.), but when we analyse these closely we will find that they don't line up identically to the Westerly notion of "a soul." Certainly, they are theories about a part of the impermanent human being that is, in fact, immortal and ageless, but these diverse theories are not all identical and not all describing "the same thing."

Instead of related to "sin (against God)," Buddhist rebirth/reincarnation (it's all the same words in Buddhism and Buddhism natively does not differentiate between "rebirth" and "reincarnation," those both being English-specific terms) is related to "karma," both the old karma which is a burden and the new karma which will propel further births.

There is no soul in Buddhism and it is not the same mind that persists going from one life to another.

It's as if one was to light a fire from another fire, the two fires are not technically the same fire but one is a cause & requisite condition for another and in that sense they are a linked and are in a sense "a fire".

The mental truths are such as feelings, thoughts, perception, things that aren't measured in mass. These are also with a cause, this or that mentality is linked to previous development and one's mentality does not become extinguished if there are causes for it's maintenance as fuel.

Buddha taught how to remove the causes for the arising of mentality and the truth & reality of it's cessation principle.

There's not an entity or being that gets reincarnated. That's a misunderstanding of reality and Buddhism.