The Dream World is a new aspect of the games, shown as part of a website called the Pokémon Global Link. The Dream World is accessed by sending your Pokémon using the Game Sync option of the C Gear. From here, you use the Dream Smoke obtained by Musharna and sync your game up to your Pokémon Global Link account located on the official site;
So basically after you do the little errand for Prof. Fennel you have to make an Account on Pokemon Global Link and use the Game Sync Function on the C-Gear to send a Pokemon from the PC to your computer. There are lot's of videos of it on Youtube.
Hope this helps.