They notice how you act. They notice how you treat people. They even notice what you wear.
The way you present yourself communicates how much you care about what you are doing.
Teachers are professionals. Dressing like one not only shows that you are serious about your job, but it can also get you in a professional mindset.
Dressing too casually communicates that you aren't all that serious about what you are doing.
A suit and tie may not be necessary, but putting a little effort into what you are wearing shows that you take teaching seriously. That attitude can wear off on the students as well.
Paying attention to your wardrobe may not be the most important thing that you do today, but it does matter. Sometimes the small things can make a difference.