I tried to go through Birkhoff and Lewis many years ago but it is not easy because they use different variables and the style is so different to modern proofs.

In modern terms, the key idea is that if a graph contains a vertex $v$ of degree k, then you can get an expression of the form $$P(G,\lambda) = (\lambda-k) P(G-v,\lambda) + \text{other terms}$$ where the other terms are all chromatic polynomials of minors of $G$.

So if the inductive hypothesis is that $P(G,x)> 0$ for all $x\geq 5$, then this reduction gives the inductive step.

Then we know that any planar graph has a vertex of degree at most 5, and that a minor of a planar graph is planar, and so we're done.

The expression involving $(\lambda-k)$ was proved by Thomassen, separately by Woodall, and more generally for all matroids by Oxley whose result preceded the others by two decades. However, although he proved the result, Oxley didn't actually state it explicitly, so missing it can be excused.

I'll dig out the BL paper tomorrow and see if their approach was essentially the same.

I tried to go through Birkhoff and Lewis many years ago but it is not easy because they use different variables and the style is so different to modern proofs.

In modern terms, the key idea is that if a graph contains a vertex $v$ of degree k, then you can get an expression of the form $$P(G,\lambda) = (\lambda-k) P(G-v,\lambda) + \text{other terms}$$ where the other terms are all chromatic polynomials of minors of $G$.

So if the inductive hypothesis is that $P(G,x)> 0$ for all $x\geq 5$, then this reduction gives the inductive step.

Then we know that any planar graph has a vertex of degree at most 5, and that a minor of a planar graph is planar, and so we're done.

The expression involving $(\lambda-k)$ was proved by Thomassen, separately by Woodall, and more generally for all matroids by Oxley whose result preceded the others by two decades. However, although he proved the result, Oxley didn't actually state it explicitly, so missing it can be excused.

I'll dig out the BL paper tomorrow and see if their approach was essentially the same.

Added I think that the result is contained in Theorem 3.1 on page 413 in the BL paper (freely available from AMS), which bounds the coefficients of a polynomial $Q$ (the chromatic polynomial after a few trivial factors are removed). They helpfully remark that in this chapter, their usage of the symbol $\ll$ is "contrary to the notation of the preceding chapter", presumably to prevent any complacency on the part of the reader.