When it's working properly, a dipole is a ' balanced' radiator, with currents of equal amplitude flowing in each leg. When you design an antenna with dipoles in it, you have to assume that the only currents that will flow will be in the antenna elements themselves. That means you don't want any current flowing down the outside of the (coax ) feeder because those currents would produce radiation and interfere with the desired pattern and impedance. There will, of course, be equal and opposite currents flowing in the inner conductor and the inside of the screen. The current flowing along the inner can split, with some flowing in the 'bottom' dipole element and some current flowing down the outside of the feeder (Kirchoff's first law). A Bal-un will ensure that current will only flow in the legs of the dipoles, eliminating the 'unbalanced' flow of current. It converts the Unbalanced situation in the coax to a Balanced situation in the antenna. At the same time, some balun designs also give you an impedance transformation between the 50Ω coax and the higher impedance of the dipole array. (Other designs exist and the same thing applies to them as well). BTW, one good way to eliminate current flowing back down the outside of the feeder is to wrap several turns of the feeder through a ferrite ring - cheap and cheerful and it doesn't help with the match.
There are unbalanced radiators - e.g. a whip antenna on the ground or on a (big enough) ground plane. Current will flow in the whip and also along the ground. The ground currents produce an 'image' of the whip that is upside down and below the ground (same as an optical image in a mirror).
Why use coax to feed a dipole system? It's convenient, already screened and cheap to buy and to install. Also, most amplifiers have an unbalanced output (Signal and Earth). You can get balanced feeder and it is used, for instance, at HF transmitting stations, where it can run across the aerial field as two open wires or as a two wires inside metal trunking. That's hideously expensive but reduces spurious radiation from bends and switches. The twisted pairs, used for Ethernet etc, are balanced, too. Twisting the wires will reduce the radiation considerably (hence low crosstalk between adjacent lines).
