Here's my explanation of the ZKP:

First, both you and the oracle know what the uncolored map looks like, and what the 3 colors are.

Step 1: The oracle sends the entire 3-colored map, but asymmetrically encrypted. So you can't see the map, but the oracle can't "un-send" it. If no 3-coloration of the map exists, the oracle has to send you something, and because you know what the map and colors look like, the only thing they can send is a map where some two adjacent regions have the same color.

Steps 2&3: You randomly pick two adjacent regions and ask the oracle for their decryption keys, so you can see their colors. When you initially received the encrypted map, you could not know whether it had two same-colored adjacent regions. But, if you happen to randomly choose the same-colored regions, the oracle has no way of not telling you. If the oracle refuses to send the decryption keys for those regions, or sends ones that don't successfully decrypt, you'll know something is up, and can assume the regions are the same color. And the only keys that successfully decrypt the regions' colors, decrypt into their original colors; so the regions will only decrypt into different colors, if they were different colors in the encrypted map the oracle originally sent.

To further illuminate: the "random pick after the map is received" ensures that the oracle must send you a map where all adjacent regions have different colors, even though you can't see all the colors yourself. Otherwise, the oracle can't guarantee that you won't ask for the two regions with different colors (because they sent the map before you randomly pick), and if you do ask for those regions, they can't respond in a way that re-assures you the colors are different (because the only way to do so is send keys that decrypt the regions into separate colors, and since they decrypt into the same color, such keys don't exist).

Step 4: Repeat steps 1-3 an unbounded number of times. This is necessary because in a single iteration, there's a chance that the oracle sends a map with two adjacent same-colored regions, but you pick two different regions; so the map is un-3-colorable, but you don't find out. In fact, this is a very high chance if it's a large map and only a single pair of adjacent regions have the same color. But more iterations increase the probability that you do find out indefinitely; with enough iterations, the probability that one of them you get lucky and select the same-colored region is 99.999...%.

Also, each time you repeat steps 1-3, the oracle sends the map with a different coloration. Otherwise, you'd slowly reveal the colors to get the fully-colored map, so it wouldn't be a zero-knowledge proof (two colors in each of two maps with different colors doesn't give you any more information than two colors in one map, so even with unbounded iterations the original coloration isn't revealed). If the map isn't 3-colorable, the randomization doesn't affect the probability: when the oracle randomizes the map, they could choose an entirely different coloration and give two different adjacent regions the same color, but the probability of you randomly choosing those two regions stays the same, so the probability of "getting lucky" in one of enough iterations also stays the same, at 99.999...%.