Because the Boyle temperature is *defined* to be where Z = 1, that's all. You might as well ask why 0C is where ice freezes -- becaus that's how 0C was defined, that's all.
What you might be asking is why such a temperature exists at all. Why isn't Z > 1 or Z < 1 always? The answer to that is that the repulsive forces that would tend to make Z > 1 have no important temperature dependence. They are there regardless of the temperature, and don't change much, because they come essentially from the fact that two bits of matter can't occupy the same space. That is, particles collide. Nothing much happens when the temperature is higher or lower, except that particles collide harder.
Not so for the atractive forces that tend to make Z < 1. These forces become much more important at lower temperatures. This is because attractive forces have a finite range and finite strength. They're like a little pool of molasses surrounding each atom: if another atom runs through that pool, it slows down as it struggle through the molasses.
The degree to which these attractive forces matter is determined, roughly speaking, by how long, on average, particles spend running through molasses. If they are traveling slowly (temperature is low) they spend a lot more time dragging themselves through molasses, and the pressure will be significantly lowered. But if they are traveling quite fast (T is high) they will fly through the molasses very quickly, and it will have very little effect on the pressure.
So, in summary, at very high temperatures, repulsive forces must dominate, and Z > 1. At very low temperatures, attractive forces must dominate, and Z < 1 (we know at a low enough temperature the atoms or molecules will simply stick together). Since Z > 1 at some high T and Z < 1 at some low T, it follows logically there must be some intermediate T at which Z = 1 exactly. We define this as the Boyle temperature.
I should add that for a model system in which there *are* no attractive forces, Z > 1 always and the Boyle temperature is 0. There are no real systems of this nature, although things like He come close.