Below is what the Rebbe said we can learn from the game.
From the words of the Rebbe: The king is the most valuable piece on the chessboard. Protecting the king and attacking the pieces which threaten the king's "dominion" is the objective of the game, and the goal of all the pieces at the king's disposal.
The same thing is true with all of created reality. The king represents the King of the Universe. When G‑d created the world, He had an end-goal in mind – that this G‑d-denying reality be made into a place where His dominion is known. Just as all of the pieces in the chess game exist only to protect the king and further his goal, all components of creation only exists in order to fulfill this deepest desire of the King of kings.
While the king represents the transcendent quality of G‑d, the queen represents malchut d'Atzilut, G‑d's immanent quality. This quality of G‑d generates the rest of the spiritual hierarchy, including all the angels and souls.
The officers -- rooks, bishops and knights -- represent the angels. They inhabit the spiritual worlds and channel Divine energy to the worlds below and are imbued with great powers.
And on the lowest rung are the pawns, which represent the souls of Jews as they are embodied in physical bodies in this world.
Every level of this hierarchy has a unique position and method of moving, in accordance with its mission.
On the lowest rung, but on the front lines, are the pawns. Like the pawn that can only go forward one step at a time, we make the world into a place where G‑d can feel at home by moving slowly, step-by-step. We do our work with simple actions that are often not very glamorous. Although we can achieve a lot, we must work within the limits of the natural universe.
However, when a pawn finally completes its step-by-step progression and reaches the other side, it can be swapped and promoted to a higher level. It is even possible for a pawn to attain the level of queen.
This is also true spiritually: It is possible for a simple human soul to be united with its source in malchut d'Atzilut, to be charged with the level of G‑dliness that is higher than all the angels and souls. We are the only ones in all the realms of created reality that are capable of this kind of drastic transformation.
This is in contrast to the officers: the rook, bishop, or horse. They can hop and skip, several steps at a time. Yet they can each only move in the way they have been assigned. The rooks only move in straight lines, the bishops only move diagonally, and the knights only go two-squares-vertically, one-square-horizontally, or vice versa.
In the spiritual worlds, each angel has its own unique character and method of transmitting the Divine flow to the lower worlds. But while angels are supernatural spiritual forces, they can "hop and skip," they are limited by their own job-descriptions. Unlike humans, angels cannot act out of character, upgrade or improve themselves.
The queen has more power and freedom than any of the officers; she can move infinitely in any direction. But freedom implies risk, and the queen is often thrown into harm's way for the sake of the game. Paralleling this, G‑d allows an aspect of Himself to go into exile, to become embedded in a world that will not necessarily recognize His presence. G‑dliness can be found everywhere and at any time, even in situations that appear foreign to G‑d.
Interestingly, the king, the most important piece, seems to have the least power. While it can move in any direction, it can only move one step at a time, like a lowly pawn. It does not engage in the fighting, and it moves only when it is most necessary, to win the whole game or in a time of danger.
This is because the king represents the innermost essence of G‑d which is completely removed from the mundane world. This aspect of G‑d does not ordinarily become engaged in the happenings of the world. But in a stunning move of extravagance, when the battle becomes a battle of life-and-death, when the whole purpose of creation is at stake, the King of kings, "G‑d" in the most infinite sense, steps in and joins us. We are never far removed even from that most transcendent aspect of G‑d.
And what does it mean to win a game of chess? What is the future that even G‑d Himself will drop everything to save? It means to win the war of all wars: when the world will be a place of good and harmony, peace and tranquility; when no part of G‑d will be in exile; and when the essence of G‑d will no longer be "removed" from creation.