Why do we introduce the idea of manifold in GR books? After reading Timaeus answer here: https://math.stackexchange.com/q/1302672/, I got an idea that spacetime we usually talk about in GR can be described as a manifold.

Firstly, let's address coordinates, how to switch and why in General Relativity you sometimes are forced to switch coordinates. We will start with a single coordinate system. For an event $m$ (a point in your manifold) in a region of spacetime $M_i⊂M$, where $M$ is the total spacetime, then there can be a coordinate map $ϕ_i$ which is a one-to-one mapping from all of $M_i$ to $\mathbb{R}^4$.

The fact that they mentioned in the last sentence

one-to-one mapping from all of $M_i$ to $\mathbb{R}^4$

means that a subsection of spacetime $M$ can be seen to be have Euclidean topology locally, even though we know that spacetime in GR is Lorentzian.
Even if we disregarded Timaeus answer there, we all have read GR books that started with talking about Manifolds and how manifolds can be defined locally as Euclidean. Afterwards, authors start to define Vectors, (covariant and contravariant), tensors and so on... thus preparing the notions one use in GR. My question is:
Why do we need to introduce students to Manifolds before teaching them about GR, if the latter has Lorentzian signature meanwhile the former is locally described as Euclidean?
 A: Space-time is not simply a manifold, but a differentiable and pseudo-Riemannian manifold.
The topology of the manifold is already fixed by the (topological) manifold structure (that is: local homeomorphism to $\mathbb R^n$ without assuming any inner product). In this sense sufficiently small open sets from the manifold carry the topology of a subset of the Euclidean space. But this construction does not define a metric on the manifold (even the notion of differentiation requires more structure, namely a differentiable structure which is an atlas, where you require that the transition maps between charts are differentiable).
While a Riemannian structure allows one to define a topology (as the Riemannian structure induces a metric, which induces a topology) this topology is always consistent with the topology of the manifold generated by the local homeomorphisms to $\mathbb R^n$.
A pseudo-Riemannian structure, which the manifold carries in general relativity, does not allow to define a topology (as the distance function is no longer a metric, as it is not positive definite). This metric does, however, allow the construction of a geometry, and a connection (or "covariant derivative") $\Gamma^\mu_{\phantom{\mu}\nu\kappa}$ which in turn leads to geometric invariants (as the Riemann curvature $R^\mu_{\phantom{\mu}\nu\kappa\lambda}$) and geometric objects (such as geodesics).
Note, that the dynamical field in GR is not the topology of the manifold, but the pseudo-metric $g_{\nu\nu}$, which lives on the manifold. So in a sense the topological manifold is a given background, on which the metric evolves. (Note, that there is some interplay, as the curvature can constrain the global structure of the manifold. There are even manifolds that cannot be given a pseudo-Riemannian metric).
In conclusion: You have to keep the notions apart. The property of $\mathbb R^n$ that is modelled by a topological manifold is only its topology not its metric or geometry. To define a metric and geometry the manifold has to be equipped with extra structure. This structure can lead to a different local geometry from Euclidean space.
