The crux of Karma Yoga by Swami Vivekananda is the concept of selfless action as a means to spiritual liberation. It teaches that every action, when performed without attachment to its results, becomes a path to self-realization and inner peace. The key principles are:

1. Selfless Service: True Karma Yoga is about serving others and performing one's duties without personal gain, ego, or desire for rewards. Through selfless service, one purifies the mind and reduces the influence of the ego.
2. Detachment from Results: The book emphasizes that we should act without attachment to the outcomes of our actions. The results of our actions are beyond our control, so focusing solely on the action itself, without concern for success or failure, leads to inner peace.
3. Purification of the Mind: Selfless actions help in reducing desires and attachments, which are the root causes of suffering and ignorance. Over time, this leads to mental clarity and spiritual growth.
4. Union with the Divine: By acting in the world with a sense of detachment, one comes closer to realizing the oneness of the individual soul (Atman) with the universal consciousness (Brahman). Through this, we achieve spiritual freedom and liberation (Moksha).

In essence, Karma Yoga teaches that spiritual growth can be achieved not by renouncing the world, but by fully engaging with it through selfless action, focusing on the process rather than the rewards.