Well, the canonical answer is that the balun converts the dipole (BAlanced) to coax (UNbalanced). But what does that mean?

For transmission lines (twin-lead or coax) to not radiate, each conductor must carry equal and opposite currents. It's these equal and opposite currents canceling each other that results in zero net field away from the transmission line. If the currents are not equal and opposite, then they don't cancel, and the difference results in an external electromagnetic field, and your transmission line radiates like an antenna.

This is generally undesirable. When transmitting, it can mean high RF in the shack, messing with speakers, digital electronics, etc. If attached to a directional antenna, it causes problems transmitting and receiving because the actual radiation pattern is a combination of the intended antenna, and the unintentional antenna formed by the feedline, which probably isn't directional at all. If you've modeled your antenna, the model assumes a feedline that does not radiate.

OK fine, you need equal and opposite currents in coax to make it not radiate. If you attach it to a dipole, you should get that. It looks pretty symmetrical, right?

Well yes, the dipole is symmetrical, but the coax isn't. Usually, the shield is attached to ground somewhere. Even if it isn't, the geometry of the shield and the center conductor are not identical, so the shield will have a different impedance with respect to the dipole than the center conductor.

Regardless of the coax, the dipole may not be really symmetrical anyway. Consider not just the antenna, but everything in its near field. Is it perfectly level to the ground? Is one end closer to a metal gutter?

The result of all these asymmetries is that each side of the dipole presents different impedances to the feedpoint. Ideally, the antenna would accept all the power sent to it by the transmitter. The equal and opposite currents put into the coax by the transmitter each go down opposite legs of the dipole where the EM energy is radiated away. But when the dipole is not balanced, the currents accepted by each leg of the dipole can't be equal. But, all the current must go somewhere, so the difference in current between the dipole legs is reflected back down the coax, back at the transmitter. However, this current doesn't have an equal and opposite current on the center conductor, so it radiates off the coax, like an antenna.

If you operate a dipole fed with coax without a balun, this is happening to you, though you may not notice. The coax shield radiates just fine, and if your signal still finds its way to the other station and you don't give yourself RF burns, then there's no problem. A good RF ground at the feedline's entrance to the shack can reduce the RF re-entry. Even without any special attention to grounding, at 100W, the RF in the shack probably won't do more than make some speakers buzz, which might be considered "normal".

By adding a balun, you are avoiding this issue. By isolating the feedline, you won't get RF in the shack, and your dipole will behave like a dipole described in textbooks. Your antenna tuning and performance won't be altered by other objects near the coax, because the coax won't be part of the antenna. If your dipole is in a good location for an antenna and your coax isn't, then you will have overall a better antenna.

Well, the canonical answer is that the balun converts the dipole (BAlanced) to coax (UNbalanced). But what does that mean?

For transmission lines (twin-lead or coax) to not radiate, each conductor must carry equal and opposite currents. It's these equal and opposite currents canceling each other that results in zero net field away from the transmission line. If the currents are not equal and opposite, then they don't cancel, and the difference results in an external electromagnetic field, and your transmission line radiates like an antenna.

This is generally undesirable. When transmitting, it can mean high RF in the shack, messing with speakers, digital electronics, etc. If attached to a directional antenna, it causes problems transmitting and receiving because the actual radiation pattern is a combination of the intended antenna, and the unintentional antenna formed by the feedline, which probably isn't directional at all. If you've modeled your antenna, the model assumes a feedline that does not radiate.

OK fine, you need equal and opposite currents in coax to make it not radiate. If you attach it to a dipole, you should get that. It looks pretty symmetrical, right?

Well yes, the dipole is symmetrical, but the coax isn't. Usually, the shield is attached to ground somewhere. Even if it isn't, the geometry of the shield and the center conductor are not identical, so the shield will have a different impedance with respect to the dipole than the center conductor.

Regardless of the coax, the dipole may not be really symmetrical anyway. Consider not just the antenna, but everything in its near field. Is it perfectly level to the ground? Is one end closer to a metal gutter?

The result of all these asymmetries is that each side of the dipole presents different impedances to the feedpoint. Ideally, the antenna would accept all the power sent to it by the transmitter. The equal and opposite currents put into the coax by the transmitter each go down opposite legs of the dipole where the EM energy is radiated away. But when the dipole is not balanced, the currents accepted by each leg of the dipole can't be equal. But, all the current must go somewhere, so the difference in current between the dipole legs is reflected back down the coax, back at the transmitter. (Note, this is distinct from reflection from impedance mismatches that causes high SWR.) However, this current doesn't have an equal and opposite current on the center conductor, so it radiates off the coax, like an antenna.

If you operate a dipole fed with coax without a balun, this is happening to you, though you may not notice. The coax shield radiates just fine, and if your signal still finds its way to the other station and you don't give yourself RF burns, then there's no problem. A good RF ground at the feedline's entrance to the shack can reduce the RF re-entry. Even without any special attention to grounding, at 100W, the RF in the shack probably won't do more than make some speakers buzz, which might be considered "normal".

By adding a balun, you are avoiding this issue. By isolating the feedline, you won't get RF in the shack, and your dipole will behave like a dipole described in textbooks. Your antenna tuning and performance won't be altered by other objects near the coax, because the coax won't be part of the antenna. If your dipole is in a good location for an antenna and your coax isn't, then you will have overall a better antenna.