What's the sense in a Hyperelliptic Riemann Surface? Can someone explain me, possibly using some very intuitive ideas, of what kind of object a hyperelliptic Riemann Surface is? What's the goal of constructing it (my lecture on is was based in Miranda's "Algebraic Curves and Riemann Surfaces")? Why is it an interesting construction?

I would suggest to read the book Farkas & Kra "Riemann Surfaces". There are many computations involving hypelliptic curves. On the other side, Miranda's book is well done and uses quite a friendly notation.
In any case, I would see the hyperelliptic curves as natural generalization of the elliptic ones, simply having a look at their algebraic definition $via$ polynomials.
One can see the elliptic integrals as motivation for elliptic curves (historically elliptic functions were discovered inverting elliptic integrals); similarly you could consider the Schwarz Christoffel mapping as motivation for the introduction of hyperelliptic curves.
From the topological point of view, they are also well studied and quite natural. I refer to Farkas and Kra's book for some examples (you should consider hyperelliptic curves with genus $g>1$ and polynomial with distinct roots to have a good visualization of the curve).
In applications you could be interested in arithmetic computations leading to hyperelliptic cryptography, or to the study of Riemann theta functions on hyperelliptic surfaces, with specific choices of the branching points. In algebraic geometry with moduli spaces or in quantum field theory there is also much to do with hyperelliptic curves.
