Intuition behind CW complexes

It's not always possible to visualize things perfectly! But in this case, the spaces involved are quite reasonable. A useful fact here is that a CW pair $(X,A)$ yields a quotient space $X/A$ which inherits a cell structure from $X$ (p.8 in Hatcher). The cells of $X/A$ are the cells of $X -A$ plus one new $0$-cell, the image of $A$ in $X/A$. You get the attaching maps by composing the original attaching maps with the quotient map.

So to get a cell structure on $S^2/A$, where $A$ consists of the north and south poles, we may choose to view $S^2$ as a CW-complex with $0$-skeleton $A$, two distinct arcs connecting those two points, and two discs attached to the resulting circle. In other words, the $0$-skeleton sits inside the $1$-skeleton as the equator, and the $1$-skeleton sits inside $S^2$ as the equator (and this equator passes through the north and south poles).

With this setup, $S^2/A$ has one $0$-cell corresponding to $A$, two $1$-cells and two $2$-cells. The $1$-skeleton is a wedge of two circles, and each disk is attached to this space via a loop $\gamma$ that encircles both circles of the wedge exactly once. Moreover, for the computation of the $\pi_1$ we may choose the basepoint to be the point where the circles meet, i.e., the $0$-cell. By Proposition 1.26 in Hatcher, we have $\pi_1(S^2/A) = \pi_1(S^1 \vee S^1)/N$, where $N$ is the normal subgroup generated by the loop $\gamma$. But we know that $\pi_1(S^1 \vee S^1) \cong \mathbb{Z} * \mathbb{Z} \cong \langle a,b \rangle$. Therefore $\pi_1(S^2/A) \cong \langle a,b \mid ab\rangle \cong \mathbb{Z}$.

CW-complexes are a generalization of the much more readily visualized simplicial complexes: https://en.wikipedia.org/wiki/Simplicial_complex. Besides purely topological research topics involving these complexes (e.g., homotopy theory), simplicial complexes are used in many other ways. One way not mentioned in the above Wikipedia article is the Finite Element Method, aimed at solving a PDE in a domain and requires first approximating that domain by a simplicial complex. Other numerical methods (e.g., the Ordered Upwind Methods by Vladimirsky et. al.) have a similar requirement. The 2-D case of a simplicial complex is a triangulated mesh. It is not a partition: the constituend triangles (simplices) share faces and vertices.