"The year does not start in August."
Translation:Nem augusztusban kezdődik az év.
Kezdi is the definite form of kezd - to start, so "kezdi az év" means "The year is starting it." What is it starting?
You need a reflexive verb here. The year itself is beginning. And that reflexive form is kezdődik.
It sounds like "When it is August, the year does/will not start."
Instead, we are trying to negate August itself, saying "not in August" - "nem augusztusban".