monkeys and ponies

You're close! The story hinges on the fact that we don't know how the Master spent all those years on Earth, or how he found the Doctor again in Spyfall II. So the story assumes that a) he somehow got into New York, b) the timeline fuckery there messed with his head, knocking him out and ultimately erasing all of his memories of what happened *except* that his future self drops him back on Earth, presumably near the Doctor, leading to c) the future Master knowing he has to rescue his past self.