The word native has to do with where you're from. You're native to the country where you were born, and places have native plants and animals too.
Things that are native are indigenous — they were born there. This is where the term Native Americans comes from — they were on this land before Europeans came over. Native plants were originally in a place — they weren't transplanted. The same is true of native animals. Sometimes people born in a place are called natives, as in "You are a native of Cleveland." When you see the word native, think "born there."