Something is flat when it is lacking in the characteristics that make it useful/interesting/unique. As noted above, this applies to soda without carbonation, women with small tits, and Kansas, but also to characters in literature or movies with little personality, tires without air, and small animals that have been run over.

As a side note, the only soda that tastes good after it is flat is Dr. Pepper, or any one of the Dr. Pepper imitations.