Very simply, the Sorting Hat—the old hat that once belonged to Godric Gryffindor that has sorted Hogwarts pupils into different Houses since the school's founding—represents Hogwarts itself, and specifically, the ideal of the diverse school that Dumbledore promoted. The act of sorting students into four different houses suggests that the founders valued difference and diversity, and they recognized that making space for all types of students would be the best way to create a welcoming environment and most effectively educate students. Voldemort, on the other hand, sees diversity as a threat to the monolithic and pureblooded society he wants to create. For him, then, destroying the Sorting Hat itself is the easiest way to destroy Dumbledore's vision of Hogwarts and replace it with his own.
The timeline below shows where the symbol The Sorting Hat appears in Harry Potter and the Deathly Hallows. The colored dots and icons indicate which themes are associated with that appearance.