What specific aspects show that this movie was set during the Great Depression?

One scene that shows that this story happens during the Great Depression occurs about fourteen minutes into the movie. In this scene Atticus is tucking Scout into bed. She asks to see his pocket watch, and he explains that one day the pocket watch will be given to Jem. Scout asks what she will get one day. He responds,

"I don't know that I have much else of value that belongs to me. But there's a pearl necklace, there's a ring that belonged to your mother. And I've put them away, and they're to be yours."

Atticus, though a lawyer, does not have much money; he has very few valuable possessions because this story takes place during the Great Depression.
Another scene occurs at approximately three minutes and twenty seconds. The narrator (an adult Scout) explains that in the town of Maycomb, Alabama, in 1932 there was

no hurry, for there was nowhere to go and nothing to buy and no money to buy it with.

The townspeople are very poor because the story happens during the Great Depression. People have to entertain themselves without spending money.
As she explains this, the focus turns away from the narrating voice and to a young actress who is playing six-year-old Scout Finch. Mr. Cunningham approaches her house trying to quietly leave some hickory nuts as repayment for Atticus Finch (her father). Again, this shows that farmers have very little money, since he has no way to offer Atticus payment other than produce from his crops. Mr. Cunningham greets Atticus politely and then leaves; Atticus realizes that Mr. Cunningham is embarrassed that he has to pay for his legal work with goods (rather than money), and he asks Scout not to fetch him next time Mr. Cunningham comes by. Scout asks if Mr. Cunningham is poor; Atticus says he is. She then asks if their own family is poor. This time, Atticus says that they are, but not as poor as the Cunningham family. He explains that the Cunninghams are farmers and that

"the crash hit them the hardest."

This reference to the Stock Market crash of 1929 (The Wall Street Crash) again shows that the movie is set during the Great Depression.