It's easier to see what's going on with a bit of rearrangement. Imagine this built of tubing:
The feedpoint is still where it would be on a dipole without the balun. The feedpoint sees the dipole as usual (blue).
In parallel with that is a twin-lead transmission line formed by the two parallel sections of the balun (red). This is a quarter-wave section with a short at the end, so it looks like an infinite impedance. An infinite impedance in parallel with the dipole at the feedpoint doesn't change the impedance the feedpoint sees. The red section might as well not be there at all.
Then, there's this green stub on the bottom. It's insignificant to the operation of the antenna because there can be no currents on it. Although there may be currents of the balun (red section), they are equal and opposite, and so at the point where the green stub is, they cancel. So you can't drive a current on the green stub via the feedpoint. Furthermore, because the green stub is symmetrical with respect to the dipole, it will have equal capacitance to either half of the dipole, and so you can't drive a current on the green stub capacitively, either.
So there's no way to drive a current on the green stub via the feedpoint. That sounds like exactly what we want in a balun, if the green stub is the coax shield. Trouble is, the feedpoint is up there at the top. How can the shield be connected at the bottom while the feedpoint is at the top? Easy, if it's made from tubing. Just cut a little hole near the feedpoint, and run the feedline inside the tubing. The shield is connected to one half of the dipole at the hole, and the center conductor goes through the hole to feed the other side:
This is equivalent to the more usual construction:
This uses the coax as the tubing, and since the spacing between the balun halves is usually very small (assuming coax diameters far smaller than the wavelength), not exiting exactly in the center has a negligible impact on performance.
By running the feedline "inside" the antenna and balun, we have the option of connecting the shield in a location where there can be no common mode current, while connecting the feedpoint somewhere else. A similar trick is used by the shielded loop antenna.