A bread box will keep bread fresher longer and will stop mold growth for at least a little while.
The bread in the bread box will still mold if you don't use up the bread in time.
The main job of a bread box is to keep the bread fresher longer and to help keep the bread from drying out as fast due to the bread box keeping moisture in the bread.
When bread is not sealed up and loses it's moisture it dries out and becomes hard and stale so by keeping the bread in a bread box the bread will stay moist longer and last a bit longer.
The mold will still grow on the bread though even in a bread box so be sure to eat the bread within a week at least.
I never keep bread longer than a week anyway because in our family we eat a lot of bread so the bread gets used before it has time to mold.
But I do keep the bread in a bread box to help it stay fresher longer.