First, let me mention the nice writing of Terence Tao on the subject: What's New - Free Probability. It best describes the spirit of the construction of noncommutative probability theory as a further generalization of classical probability theory by focusing on, instead of events and the sample space, the algebra of random variables and their expectations.
It would be an abundance to revisit everything from the foundation of the theory since there are already extensive resources in the literature. The most comprehensive text is due to Mingo and Speicher [MS17]. A good place of information is Roland Speicher's website.
In this post, I will just write down what I want to know and would be useful to my research.
[MS17] Mingo, J., Speicher, R., Free probability and Random matrices, Fields Institute Monographs, Springer, Press, 2017.