What does Miss Maudie mean when she says, "I think Atticus put his gun down when he realized that God has given an unfair advantage over most living things"?

It can be reasonably inferred from Ms. Maudie's statement, "I think Atticus put his gun down when he realized that God has given an unfair advantage over most living things" that not only is Atticus a skilled marksman, but that he possesses an abiding sense of fairness, and compassion for those who are disadvantaged, whether economically, socially, or physically.
Typically, in youth there is the journey to know oneself. We test our mettle against appropriate challenges in order to grow and learn. Atticus, in his youth, realized two things: 1) he was a gifted shot, and yet, 2) killing something that cannot defend itself was not so much a principled attribute to possess, but an empty victory without meaning.
Rather, Ms. Maudies statement supports the notion that Atticus learned about the substantial principles of humanity such as compassion and empathy, via his ability as a crack shot, only to discover he did not wish to be measured by such means.

Miss Maudie is referring to Atticus's skill with a rifle. For many years, he's had a reputation as a crack shot, earning the nickname "Ol' One Shot." But because Atticus is such a modest man, he never told his children about it. However, Atticus is reluctantly forced to display his special skill when he's called upon to shoot dead the rabid dog Tim Johnson.
Miss Maudie is offering a theory as to why Atticus put his gun down all those years ago. She thinks that Atticus realized that being such a skilled marksman gave him a terrible power over living things, an unfair advantage over them. If he wanted to, he could kill just about anything. But he's chosen not to; he'll only get out his gun if he really must, as in the case of Tim Johnson.