Since you already have a proof for $W(3,4)$ and you say you have read the GRS book, I am going to skip any discussion of how the proof actually works, and just focus on the part you say you are missing, which is the induction. This is actually a triple induction, first on the number of colors $c$ for fixed progression length $n$, then on the progression length $n$
Then for particular $(c, n)$ pair, we inductively find, for each $i$, a structure which contains a single "target$^i$" integer which would necessarily completes a progression if it is any of $i$ different colors. Once $i=c$ we are done.
The base cases are that $W(2, c) = c+1$, and $W(n, 1) = 1$, even easier.
Now we want to find a bound for $W(n, c)$. We can assume $n>2$, $c>1$, and we know some $W(n', c')$ for all pairs $(n', c') < (n, c)$, which means that either $n' or ($n'=n$ and $c').
I have not edited the following very carefully, and I wrote it rather off-the-cuff, so there may be many errors, especially off-by-one errors. But I think the idea is sound and well-documented. I will be glad to get comments and corrections.
Consider a block $B_1$ of $b_1 = 2W(n-1, c)-1$ elements; we can be sure that the first half of this block (elements $1\ldots W(n-1,c)$) contains a length-$(n-1)$ monochromatic progression of some color $\chi_1$. If the progression is $x_1, x_2, \ldots x_{n-1}$ then $B_1$ also contains a "target element" $2x_{n-1} - x_{n-2}$. If this element is color $\chi_1$ we are done, so assume that it is some other color.
Now consider a superblock $B_2$ consisting of $b_2 = 2W(n-1, c^{b_1})-1$ blocks of size $b_1$. (I have $W(n-1, c^{b_1}$) by the induction hypothesis.) I will imagine that this is a sequence of blocks each of which is colored with one of $c^{b_1}$ colors. I can assume that the first half of this sequence (elements $1\ldots W(n-1, c^{b_1})$) contains a progression of $n-1$ identically-colored blocks. Each of these contains a "target element" which by the previous paragraph cannot be colored with color $\chi_1$; say they are all color $\chi_2$. By the construction is GRS, the target elements themselves form a monochromatic arithmetic progression of length $n-1$ and they point to a "target² element" which cannot be color $\chi_1$ or $\chi_2$. Then say that the target² element is color $\chi_3$.
Now we are going to jump to the induction step. I have an super$^{d-1}$-block consisting of $b_{d+1} = 2W(n-1, c^{b_d})-1$ blocks of size $b_d$. I can assume that the first half of this sequence (elements $1\ldots W(n-1, c^{b_d})$) contains a progression of $n-1$ identically-colored blocks. Each of these contains a "target$^{d-1}$ element" which cannot be color $\chi_1\ldots\chi_{d-1}$, and the $n-1$ target$^{d-1}$ elements form a monochromatic arithmetic progression of some color, say $\chi_d$, whose next element therefore becomes a "target$^d$ element" which cannot be color $\chi_1\ldots\chi_d$.
It suffices to apply this induction step $c$ times, so the final block is the super$^{c-1}$-block consisting of $b_c = 2W(n-1, c^{b_{c-1}})-1$ super$^{c-2}$ blocks of size $b_{c-1}$
This is enough to prove that $W(n, c)$ actually exists. But we might like to know how big it is. We have $b_{d} = 2W(n-1, c^{b_{d-1}})-1$ and $W(n, c) = \prod_1^c{b_i}$ which gives a recurrence for a bound on the size of $W$. I have to get back to work now so I will leave that for you.
I expect this to contain many errors, so I have made it community-Wiki, and I cheerfully invite everyone to edit the answer without consulting me.