An Exhaust Flexi is constructed of three layers, the inside has a interlocking inner, a middle ribbed section (similar to a Exhaust Bellow) that re-enforces the inner layer, and a outer woven stainless steel wire exterior, its purpose is to act as a point of movement in an exhaust. (these can also be used for external wastegate pipes).
They reduce the weight and vibration being transferred back up through your headers or turbo setup, this weight and vibration is created from the length of exhaust systems. If your exhaust is well mounted and supported then you will have less chance of issues. Without an Exhaust Flexi the chances will increase of bolts, studs and nuts loosening, and the cracking of exhausts, downpipes, headers & turbo manifolds.