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Probably there are many reasons why people care about foliations, but for someone coming from operator algebras one of the main reasons is the connection to von Neumann algebra theory. In brief, every foliation of a compact manifold has an associated von Neumann algebra and interesting properties of the von Neumann algebra are reflected in geometric properties of the foliation. The von Neumann algebra is a factor if and only if the foliation is ergodic, for example. You can get examples of factors of all types by this construction and the geometric aspect of foliations is especially helpful in understanding the type III case. The modular automorphism group has a straightforward geometric interpretation, and so on. A good reference is "Operator algebras and the index theorem on foliated manifolds" by H. Moriyoshi.