But how does "bug" match the meaning of "antepenultimate" in any way? I must be missing the point of the game.

Is it just "words that vaguely, extremely distantly, have some relation, as determined by the neural net"? And your job is to work out which extremely-distant word is slightly closer? (I noticed "gubi" scored very, very slightly more points than "bug" in the demo.

You are basically correct. https://en.wikipedia.org/wiki/Word2vec

Essentially, all words are embedded in an N-dimentional space. Then you can simply measure the distance between words in the word-cloud. The exact method of how it decided what coordinates to assign to each word in the cloud isn't important to the game.