Two-cocycles turned up, and were used in the study of central simple algebras before group cohomology was defined. The use of group cohomology greatly simplifies the classification of of central simple algebras over $\mathbb{Q}$, for example. Moreover, once you restate the classification (the Albert-Brauer-Hasse-Noether theorem) more abstractly in terms of group cohomology, you can apply it elsewhere.

Homology/cohomology is just like anything else in mathematics. At first it may seem strange, but once you understand it, it becomes familiar.

In algebraic number theory and class field theory, you mainly need the cohomology of finite groups, which in fact is a very good place to start, so my advice would be to first study thoroughly the cohomology of finite groups, as for example, in Part 3 of Serre's book Local Fields (Corps Locaux). From there, it is not difficult to understand the cohomology of profinite groups.

Loosely speaking, etale cohomology unites the usual cohomology of manifolds with the cohomology of the Galois groups of fields, and is of fundamental importance in arithmetic geometry.