Literally, it means they are more than 35 years old. You can think it as some kind of "age curve." You go up from 30 to 35, and at the moment of 35-year-old, your 30s begins to go downward, so the curve changes its direction. If a line on a graph suddenly changes its direction it looks like it "broke" at the point, so it "꺾이다".

OMG... I've never expected someone ask this hahahahahaha Simply 'I'm too old' or 'I'm over certain age' Literally means "I'm late 30's"