Why do vapour cones form around jet fighters?

It forms a cone because it depends on a shock wave, and the region enclosed by the shock wave appears conical in shape.

See, for example, the apparent cones here:

They are also visible here:

Wikipedia appears to be fairly clear on why vapor cones are related to shock waves. From the introduction to the article about vapor cones:

Atmospheric water then condenses, and thus becomes visible, as air pressure decreases suddenly across shock waves associated with supersonic flow speed.

It's also put nicely on the page for the Prantl-Glauert singularity:

The reason for the condensation cloud that is being observed is that humid air is entering a low-pressure region, which also reduces local density and temperature, sufficiently so to cause condensation. The vapour vanishes as soon as the pressure increases again to ambient levels.

So we simply have to figure out why these shock waves are conical. The answer to that is essentially that the region inside the cone is at a much different pressure than the region outside it. At any given point in space, the boundary cross-section expands uniformly in all directions (hence it is a circle). In the outside region, air molecules are at one pressure; in the inside region, the molecules are at another pressure. The shock wave is the boundary between these two regions. As it travels outward, the molecules it passes through become the same pressure as the other molecules inside the cone.