Alice is first seen drinking from the bottle, which shrinks her to a small size. She then does so again when she is giant size, and grows small in midair, plummets down, lands in the bottle, and starts floating through the Doorknob, where she enters Wonderland.
The bottle is one of the first magical objects that appeared by the Doorknob, the others being the table, the key, and the Eat Me Cookies.
In the book, the liquid in the bottle tasted like cherry tart, custard, pineapple, roast turkey, toffee, and hot-buttered toast; however, only the first four flavors were mentioned in the film before Alice realized something different happened to it (or better yet, to her), but as for drinking the last drop from the bottle, it is unknown whether it tasted like either of the last two or both.
Early concept art
Shrunk Alice sitting next to the magic Drink me Bottle magically appeared by the Doorknob