I've just started studying Kobayashi's Foundations of Differential geometry and I was wondering why the author introduces the concept of "pseudogroup of transformations" in the very beginning of the book. Up to now I haven't found this concept in anywhere else in mathematics, is it an useful tool? Is that concept really needed for studying differential geometry?
A book on differential geometry has to define a smooth manifold somewhere at the beginning. To define a smooth manifold, one has to explain what an atlas is. Kobayashi found a way to shorten the definition of an atlas (and make it easily adaptable to different notions of smoothness) by using the language of a pseudogroup of transformations. The crucial compatibility property of an atlas then simply says that transition maps $\varphi_j\circ \varphi_i$ belong to an appropriate pseudogroup.
Not explicitly. If you pick a random textbook on differential geometry, chances are that it will not have the term "pseudogroup of transformations" in it. But it will surely have the definition of a "smooth manifold", a part of which (the one concerning the transition maps) conveys the same information as the statement about compatibility with a semigroup in Kobayashi's text.
The term pseudogroup of transformations does come up elsewhere. But for the purposes of reading Kobayashi's book, you can probably forget it, and keep in mind only the important examples which appear at the bottom of page 1.