I was wondering why is research on grammatical gender important? Why is exploring this area of linguistics of any interest to linguists? What can it tell us about language (especially with regards to cognitive linguistics)?
There are a few million answers (32, if I'm not mistaken), here is one. Bantu languages have a complex system of grammatical gender where nouns have some gender, and things that agree with nouns agree in gender (we call them "classes"). The marker that you get depends on a lexical property of the noun, but also on whether it is singular or plural. What happens when you conjoin singular nouns and you have to agree with that conjunction? If they are the same class, it's just the "corresponding" plural form; but what if they are in different classes? Then things get very complex. In some languages, there is a hierarchy of the type "If A & B then agree in the plural of B; If C & B, then agree in the plural of C" – in other words, order the classes A < B < C and take the highest class in the conjunction. How exactly does this business of resolving differences in grammatical gender work? That's a very interesting question.
Grammatical gender is a subset in systems of noun classes, which occur in different forms in many languages. Noun classes describe how a language categorizes nouns for the purposes of counting, morphology, relating to verbs and adjectives, etc. Gender is also related to number (e.g. singular, dual, plural) and lexical classifiers (as noted in the comments).
Noun classes have implications for cognitive linguistics (see the huge literature based on Robin Lakoff's book mentioned above), but the fundamental reason why people study them is to better understand the mechanics of each language, just as scientists study the physical characteristics of animals and viruses.