Exclusive means with limited access. The only way you might get reservations at one of the most exclusive restaurants in Los Angeles is to become friends with the maitre d', or become famous.
By its nature, something that is exclusive leaves people out, or excludes them. You would think this was a bad thing, since excluding people is not very nice, but by being exclusive, things like clubs, restaurants, and resorts become all the more desirable. Exclusive can also mean sole or only: The company became the exclusive soda vendor at the new stadium. The local newspaper got an exclusive interview and broke the news.