In many religious and philosophical traditions, the soul is considered to have some form of existence independent of the body. While in some religious systems (Christianity and Islam, for example), this takes the form of a belief in the existence of the soul after the death of the body, in others, such as Hinduism and Platonism, this includes a belief in reincarnation or the transmigration of souls.
Within such belief systems, the soul existed before it descended into the body, usually in a state of some sort of union with the divine. Before the descent into the body, the soul shared some form of divine knowledge, but when souls became embodied (i.e. when people are born), they forgot their divine origins. Under this view, life is a quest to recollect and return to our divine origins, and many religious and spiritual practices are geared towards facilitating this.
Thus the phrase "remembering the soul's history" refers to remembering the soul's divine origin before it was born into a body.