I don't know why you think it's an emergent ability.

It's seeing a sequence of moves, and playing the most likely next move (i.e. the most likely next token) given the previous complete move sequences it was trained on. That's the baseline of what an LLM does—not something emergent. Games in online chess databases tend to be of relatively good players. Nobody wants to look up games played by two 800 ELO players.

As an aside, there have been chess programs for years that show you for a given position all of the previous games in its database with the same position and the win outcome % of each move. That's all that's going on here.