Technically speaking, it's because giyerim is in the aorist tense which makes it a habitual thing.
So in this case it's more like "I habitually wear shoes at the square", or "when I'm at the square, I wear shoes".
Your translation suggests that you're already at the square, wearing those shoes that you like to tell everyone that you tend to wear there!
"put on" is the most commonly used phrase for the verb "don." It's not commonly used anymore, but it specifically refers to act of attiring oneself. Sometimes you'll come across a sentence like "He donned his hat and walked away." After an article of clothing is donned, it is worn. That to say, I interpret this sentence not to mean "I put on my shoes at the square," but rather "I (continue to) wear my shoes at the square."
Does Turkish have a verb or phrase equivalent to don? My dictionaries aren't helping, they think I'm asking for "don't," and searching "put on" returns giyemek and tanımak.