Were there real historical examples of “mystery knights” competing in tourneys? If so, why would they want to compete anonymously? It doesn’t make a lot of sense to me even in ASOIAF.

Jousting in disguise was a big thing in chivalric romance, and thus trickled into the real jousting culture. Henry VIII, for example, loved to joust in disguise. 

As to why, well it boosts the drama and attention: people want to know who the mystery knight is, so there’s all this tension that gets built up and then released when you have the unveiling.