What exactly was Kempe's error in his proof of the four colour theorem?
What I understand of his general idea is by the following case: Suppose an uncoloured "country" is surrounded by countries of four different colours Red (R), Blue (B), Green(G), Yellow(Y) with R,G and B,Y not sharing borders. Either there will be a contiguous chain of countries forming a red-green chain from R to G or not. If not, we can interchange the colours in the maximal such chain initiated from R to change the R colour to G. In this case we are done for now only three colours surround the uncoloured country. Otherwise due to planarity such a contiguous blue yellow chain will not exist from B to Y following which we are done again by a similar argument.
As far as I know this argument drops when five countries surround a given uncoloured country. Exactly how that happens is not clear to me.