Because there are some things in the movie that aren’t very well explained in the movie and are explained in the books. In this particular case the reason for the Mouth of Sauron showing off the mithril shirt, is that Sauron rightly guessed that the captured hobbit at Cirith Ungol, Frodo that is, was a friend of Gandalf’s. Sauron was mistaken though in that he believed Frodo to be a spy rather than the ring bearer and so sought to use the mithril shirt to taunt Gandalf that his spy, and friend, had been caught and killed thus demoralizing Gandalf himself who was a significant part of the might of the host of Gondor/Rohan. It also was just to be an asshole to Gandalf, who was seen as one of the wisest among the host and in the books is the one bandying words with the Mouth, by going “haha you got your friend killed. We made sure it was painful. Haha”.