There's a nice overview of many of the "higher categorical issues" in John Baez's paper Higher dimensional algebra and TQFT. It's also very friendly for beginners.

For the details, from a perspective emphasising the 2- and 3-categories that make everything tick, you should read

• Bakalov-Kirillov
• Turaev's big blue book

Dan Freed's paper are also very good. For a somewhat different perspective, try Kevin Walker's TQFT notes, on his webpage.