Dimensional Arrival in the Reverse WorldEdit
Vegeta ended up in the Reverse World by using Instant Transmission right before Planet Vegeta exploded (which, also saved Leaf's life). Leaf's "kind personality" originally drove him crazy, but eventually he learned to ignore it. Apparently he knew Leaf before Planet Vegeta exploded, but it is currently unknown how.
The Fight Against ArceusEdit
This is self-explanatory.
Vegeta first appeared in the story The Fight to Save the Planet when he joined forces with Leaf and decided their first goal was to destroy Arceus. When Vegeta and Leaf learned of Arceus's horrifying power, they set their minds on another goal: preventing Planet Vegeta from exploding in the past. They failed. The force of the explosion was what caused Leaf's memory loss, and that is also the reason Vegeta started calling Leaf "Darkrai". (After all, Leaf is a Darkrai.) After The Fight to Save the Planet, Vegeta appears in every part of the series.