Reason 1: Character consolidation to avoid audience confusion and overload
Since movies typically only take 2-3 hours of your time, it can be difficult to familiarize yourself with a large number of characters the way you can while spending dozens of hours reading a book. For this reason, it is much easier to have a very large cast of characters in a book than in a movie. One way to make it a little easier on the audience when doing an adaptation is character consolidation. Since Glorfindel doesn't play much of a role in the books after the race to the ford, why confuse things with an extra character? Instead the writers took the opportunity to introduce a character who definitely had to appear later: Arwen. Using Arwen here makes sense since she lives in nearby Rivendell. They could have used Legolas (as they did in the 1978 animated film), but they wanted to show him and the dwarves and Boromir all arriving for the council of Elrond at a later point. This kind of consolidation is done in other places in the film--such as when Merry and Pippen and Sam recite lines or do things that Fatty Bolger did in the book.
Reason 2: Efficient exposition
By using Arwen here, the director gives himself an opportunity to give the audience some exposition about Arwen and her relationship with Aragorn. Through a few lines of dialogue, we learn that they know each other and seem to be romantically involved. Having her take Frodo on the horse gave her a heroic scene in a movie where female characters didn't get to do much. We also learn that she is a strong and capable character when it is she who raises the flood waters (though I think Gandalf still takes credit for the horses) and a definite good guy. This efficient use of time allows the director to get a head start expositing the main romantic sub plot of the films.
Reason 3: Stunt-related practicality
The race to the ford is the climactic action scene of the movie's second act and it is a visually exciting series of stunts involving real horses, not achieved with CGI. The book had Frodo ride alone, but this presents a problem for filmmakers. Throughout the movie, they use several methods for making the actors look hobbit-sized and none of them would work well in this scene:
1) Enlarge the sets and props like they did with Gandalf's oversized wagon for when Merry and Pippen stole fireworks. Well, a horse is not a prop, you can't enlarge it to make a stuntman look like a hobbit.
2) Use forced perspective where the hobbit actor is farther away from the camera than he looks, like when Frodo sits next to Gandalf on the wagon at the beginning of the movie. Easy to do when the hobbit is sitting near the big character. Impossible to do when the hobbit has to be sitting on top of the big horse.
3) Use a little person actor as a stand-in, like when they show the hobbits running alongside the other characters (usually with their hoods pulled up to hide the face). Potentially could work, but you would need a little person who was also qualified to do horseback stunts... likely unavailable.
4) CGI, like when the characters are shown side by side where their faces are visible and forced perspective is not possible. It would be very tough (especially in 2003) to get the CGI to look right covering up a large actor with a hobbit-sized image on a running horse.
So a fifth solution had to be found in order to show the race to the ford without Frodo looking too large or too CGI'd. That solution was to have a big character ride the horse with Frodo as a passenger. Note that this also corrects a bit of unrealism from the book--that a hobbit could competently ride a full-sized horse. Arwen was there and had the fresh horse. She made the most sense to be the rider. For the hobbit in this case, they used a dummy (see this site for a photo of the Frodo dummy they used). This is the practical reason why Frodo did not ride the horse alone as he did in the book.