Zatch lost his memories after he had already arrived on Earth. He was defeated by another mamodo, who decided to erase Zatch's memories instead of destroying his spellbook.
The mamodo is named Zeno, and he is revealed to be Zatch's twin brother.
As for his motivation: Zeno had a painful childhood, enduring brutal training on a daily basis. He grew up separately from Zatch, and believed that Zatch was living an easy, enjoyable life elsewhere. Not only that, but Zatch had inherited their father's great power, "Baou" (Zatch's 4th spell).
For these reasons, Zeno was extremely resentful of Zatch, and came to believe that Zatch was the reason his own life was so terrible.
Once they were on Earth, Zeno sought out Zatch and defeated him. Zeno decided to leave Zatch's spellbook alone, because destroying Zatch's spellbook would just send him back to their world, where he imagined Zatch would be living happily.
Instead, he stole Zatch's memories, so that Zatch would be forced to wander around, confused and alone, being constantly attacked with no idea why.
Later in the manga, Zeno accidentally looks at the memories he stole from Zatch. He realizes that Zatch's life was also painful, having suffered greatly at the hands of an abusive stepmother. Zeno regrets giving Zatch amnesia, and apologizes to him.
The circumstances of Zatch's amnesia are revealed in the chapter "Stolen Memories" (Volume 5, Chapter 48).