Instructions
Go to Pacifidlog Town. Find the Poke Center and then go to the house to the southeast of the center.
Talk to the old man in the house. He is always watching for Mirage Island, and he is the only person who can see it. As such, you have to check in with him until he says he can see Mirage Island. When he can see it, you can then go and find it.
Go to Mirage Island. To find it, go to the big, blank spot on Route 131 on the map, just before Route 130. The area is in the direction the old man is looking when he says he can see the island.
Walk through Mirage Island and wait to encounter Pokemon. Battle these Pokemon, and when their health is low try to paralyze them or put them to sleep. Once that happens, throw one of the Pokeballs at them and try to capture the Pokemon. The more powerful the Pokeball, and the weaker you've made the opposing Pokemon, the more likely you are to capture it.