The nefesh of the sinners of Israel in the era preceding the coming of the Messiah
In Kabbalistic literature, nefesh means the lower, external part of a human's divine soul.
Those who break the law of Torah
This refers to secular Zionists of the early twentieth century. See also the note to paragraph 2.
These true believers are strongly focused on the book and tradition, and their independent sense is underdeveloped.
In Kabbalistic literature, ruach means the higher, more internal part of a human's divine soul.
Because secular Jews have a healthier nefesh, the external part of the soul, they better perceive the divinity in the external world, in the life around them and the normal course of history. (This is why they became Zionists, having heard the voice of God resounding in history and calling them to Zion).
The soul (nefesh) of the first group (secular Zionists) is of a better nature than the second group, but is subjected to the powerful negative influence of a spirit of transgression, and so gradually continues to get worse.
The light of the Messiah refers to the light that emanates from the divine purpose (the coming of the Messiah), rather than from an evaluation of the ephemeral state of things. In this light, every segment of the Jewish people is regarded not according to its present level, but to the contribution it will one day bring to the future ideal. (See also paragraph 5 above.)
This refers not to "technical" or superficial study of Kabbalah (which has become popular in recent times), but to the deeper understanding the Kabbalah offers into the processes currently taking place among the Jewish people as a whole and in the State of Israel. Such an application of the Kabbalah for an understanding of historical processes in the life of the Jewish people is the Torah of the Messiah's deliverance. (Note that even in the sixteenth century, R. Itshak Luria, the Arizal, believed that the process of messianic deliverance would require a renewal of Torah, which would be linked to broader study of Kabbalah as a vital component of Judaism.)
Within this unified Jewry, each group will offer the good that it brings, and accept from another group that which it lacks.
That is, the secular Zionists, who are "sinners" in that they transgress many specific laws of Judaism (Sabbath, kashruth, and so forth), are nonetheless "good" in regard to the rebirth of the Jewish nation. Therefore, Rav Kook refers to them as "perfected sinners."
All progress that can be attained by humankind in the material and spiritual realms, in scientific knowledge, in the social sphere, in emotion, and in art (regardless of religion) is the realization of spirituality.
Each of which made possible the rectification of the other
In other words, repentance and rectification will take on not only an individual but also a national and universal character.
This term can be understood literally as "those possessed of the higher divine spirit." The word neshamah, used here, refers to the Kabbalistic idea of the third level of human's divine soul, deeper and more sublime than either nefesh or ruach. Thus, this refers to those who understand the essence of both groups discussed above, fully realizing the ideals of both - and therefore becoming the authority for each with the ability to influence its adherents.
The "perfected sinners," possessed of a rectified soul (nefesh,) are the left side of the Jewish people, whereas the God-fearing Jews are the right side; and with the aid of the "supernal tzadikim," masters of the third level (neshamah), the light of rectification of the nefesh comes to the God-fearing non-Zionists, and the light of rectification of the ruach to the secular Zionists.
The Messiah will send the light "from the future" and thus help us to rectify ourselves. Only when we look not to the past, but to the future, only when we cease to evaluate the virtues and sins of different groups, but see their future unity, can we aid in the transfer of positive influence from one group of Jews to another.
The tribe of Judah - and King David in particular - is known for its special powers of teshuvah. Striking examples of this in the Jewish tradition are the stories of Judah and Tamar (Genesis 38) and David and Bathsheba (1 Samuel 11; see commentary there and in Moed Katan 16b). Therefore, the "Messiah, son of David" has unique powers of teshuvah, and in the light of his coming, various groups of Jews will be able to rectify themselves and one another.
See Psalms 132:10. We see here an additional dimension that Rav Kook brings to our understanding of "the Messiah, son of David." This Messiah is the son of David not only in the sense that he is David's descendant, but in that he takes up the burden of teshuvah: he is able to bring all of the Jewish people to repentance and perfection, both secular and religious.