Why does the author of Twilight ask us to believe things in the novel that don't exist or that could not happen?
A good writer will craft such a enthralling tale of characters or plot that we won't care if "it couldn't really happen." As mentioned above, a reader must be willing to have a little suspension of disbelief when reading fiction, or they might as well read only non-fiction. In regards to the Twilight books, she has created characters that readers can easily identify with and cheer for. They may be vampires, but on the personal level of their values and attitudes, that doesn't matter. The vampires represent some of the traits we most admire: true love, loyalty, sacrifice, friendship, etc. Meyers has created a typical romance novel, but changed the game a bit with the vampire element to create the extra tension in the plot and adding that element of depth.
When the author's purpose in writing a piece of literature is to entertain, then we must assume that there will be a lot of literary licenses aiming to do just that for the reader.
Among the licenses that an author can opt for we have personification, the use of fables, and the application of fantasy, among many others.
Twilight deals with a trending topic among young and adult readers which resounds not only in literature but also in regular TV and movies. Therefore, the more fantastic and enjoyable the topic is, the more likely that more people will be come interested in this type of literature.
So much literature (especially stuff you're supposed to read for fun) is meant to be "escapist." It is meant to take you away from your mundane everyday world and let you live in a world of fantasy. When we read things like this, we lose ourselves and our worries and we get to simply imagine how things would be if there were vampires or witches and wizards or aliens.
Imagination is important to people. We constantly imagine things that are not or (seemingly) cannot be. This helps us have the desire to push ahead. As Browning said "oh, that a man's reach should exceed his grasp, or what's a Heaven for?"
You are right! There are many "facts" and events in the novel that certainly do not exist or could happen in real life. Most importantly, vampires themselves do not exist. So why would an author ask us to believe in their existence? In the 1800s, poet Samuel Taylor Coleridge wrote about this same issue. He said that to enjoy a piece of literature that does not deal with the real world, readers must practice "the willing suspension of disbelief." He meant that to enjoy a poem, a story, or a novel that deals in the supernatural, readers must put aside their logic and ignore for a little while their disbelief in supernatural characters and events.
Twilight is a novel about vampires, supernatural beings. The author doesn't believe in their existence either, but she has created an imaginary existence for them to tell her story. To enjoy Twilight, then, we have to put aside our disbelief in vampires and imagine for a little while that they do exist.
Just like another fiction book with vampires, this is the same. Meyer's theme was of love and in this book she included mythical creatures that no one truly believes in the real world. But this was all a part of her series and she crafted her own tale with different creatures. With the romance novel she added twists in the plot and changed it a bit to add depth.