What convinced Saber to destroy the Holy Grail in the Unlimited Blade Works movie?
In the movie, as Rin was carrying Shinji out, she told Saber to destroy the Holy Grail "cocoon" (Shinji was too weak and incompatible to be able to contain the Grail, so this "cocoon" is the result of it manifesting). 

Saber she was hesitant because Rin was still there, but she knew that if the cocoon were to explode the cursed contents would destroy everything it touched. So destroying the cocoon would be natural to her in this situation.

The movie plays out almost exactly as the visual novel scenario, with a few bits and pieces cut out for time. E.g., Saber tries to cross the pond of corrupted mud to save Rin, but Rin uses a Command Seal to prevent her from doing so (because she would be corrupted by the cursed mud if she touched it).

About the corruption of the Grail:

The contents of the Grail was corrupted during the Third Fuyuki Grail War, which allowed for the summoning of non-Heroic Spirits such as Rider and Caster. Any wish made upon this upon the Grail's corrupted contents would come with destruction. The Lesser Grail, the container meant to hold the contents of the Holy Grail was destroyed in the previous Grail War (the Fourth, by Saber, commanded by Shirou's foster father Kiritsugu using his command seal).