I'd like to know if anyone has a good explanation for where the ghost components that are used to define Witt vectors come from. A lot of sources I've read take the ghost components for their starting point.

The closest I've seen to an explanation was in Joe Rabinoff's notes on Witt vectors. This is my paraphrasing what I understood from them. Consider $p$-typical Witt vectors and let $k$ be a perfect field (or ring) of characteristic $p$. We have a multiplicative Teichmüller map $[\ ]: k \rightarrow W(k)$ which is a section to the projection map. Every element of $W(k)$ can be uniquely written as a series $$\sum_{i=0}^{\infty} p^i[a_i],$$ with $a_i \in k$. If we know how to add witt vectors, then we will know how to multiply them as well, because the Teichmüller map is multiplicative.

The series expression for $[a_0] + [b_0]$ is already not obvious. Write $\sum p^i [c_i]$ for this expression. We have $c_0 = a_0 + b_0$, but to find $c_1$ we need the following trick. Because our field is perfect, there is no harm in replacing $a_0$ and $b_0$ by $\alpha^p$ and $\beta^p$. Then we want to compute $[\alpha^p] + [\beta^p] = [\alpha]^p + [\beta]^p.$ We know $[\alpha] + [\beta] \equiv [\alpha + \beta] \mod p$. Raising both sides to the $p$th power, we can find a new expression for $[\alpha]^p + [\beta]^p \mod p^2$, and this gives us $c_1$.

The formulas that are used to find $c_1$, $c_2$, etc. look similar to the ghost polynomials. [They're a little different because the Witt vector corresponding to a series as above is $(c_0, c_1^p, c_2^{p^2}, \ldots)$, and so we haven't found the usual Witt vector components. On the other hand, the usual ghost components don't involve any $p$th roots, which is what $\alpha$ and $\beta$ are. Finally, the ghost map tells us not just how to add $[a_0]$ and $[b_0]$, but how to add any two series $\sum p^i[a_i]$ and $\sum p^i[b_i]$.]

I'm willing to believe that we can find precisely the usual ghost components this way (although I haven't actually succeeded in doing this). So it seems natural that the ghost map $w: W(k) \rightarrow k^{\mathbb{N}}$ (where the ring operations on $k^{\mathbb{N}}$ are componentwise) is an additive map. What I find surprising is that it's also multiplicative; it seems like we didn't do anything to guarantee this.

Am I correct that this is one way to find the ghost components? Is this the most natural path to them? Should it be obvious that the ghost map is multiplicative?

I'd be happy to hear how you think of the ghost components.