Why do we study closure properties of formal languages?
I think that the more fundamental question here is why study specific kinds of formal languages at all. One answer is that formal languages of specific kinds have been found useful in the construction of compilers. Another answer is that in the past they played an important part in artificial intelligence (nowadays statistical methods are more important).

Once we agree that regular languages and context-free languages are an interesting object of study, closure properties are basic mathematical properties of these objects. In fact, part of the beauty and appeal of the subject of regular languages is the large number of properties that they satisfy, including closure properties. Regular languages serve as a model for other families of languages, and one tries to mimic one's knowledge of regular languages on other families of languages.

Closure properties are also interesting beyond formal languages &mdash; Tychonoff's theorem in topology is one example.

Specific closure properties of regular languages have given rise to a new type of algebraic structure, Kleene algebras, which abstract some of the closure properties of regular languages and of context-free languages.