How is $(\mathbb{S}^2,\pi)$ a covering of $\mathbb{RP}^2$? My lecture notes state that a basic example of covering space is $(\mathbb{S}^n,\pi)$ where $\pi$ is the projection $\pi(p) = [p]$ to $\mathbb{RP}^n$ the projective space considered as the sphere with opposite points identified.

Next, I wonder what is the intuition behind this result. How does one visualize this covering? Let me explain myself: in the case of $\mathbb{R}$ and $\mathbb{S}^1$ it is easy to see that each loop of the fundamental group can be unrolled into a segment $[k,k+n]$. Is there a similar view in this case?
Response to first thing: No, it is the quotient map 
$$\pi: S^n \to S^n/\sim (i.e., \mathbb{R}P^n)$$
$$x \mapsto [x].$$
Response to second thing: I don't see how your example is an "intuition for being a covering map". But the fact analogous to your example is that every loop in the fundamental group of $\mathbb{R}P^n$ can be corresponded to half a great circle connecting two antipodal points and its concatenations.

However, I'd rather say that "the intuition"  for being a covering map coincides with its proof: it is due to the fact that it is a quotient of a properly discontinuous action of a discrete group (namely, $\mathbb{Z}/2\mathbb{Z})$.
$\mathbb{R}P^2$ is the quotient of $S^2$ endowed with the equivalent relation $xRy$ if and only if $x=y$, or$ x=-y$. So it is this quotient map. It is a cover since the restriction of the canonical projection $\pi:S^n\rightarrow \mathbb{R}^n$ to an open hemisphere is a diffeomorphism onto its image. It like so you can visualize this covering, since $x$ and $-x$ are in two different open hemispheres, so $\pi$  is a double covering map.