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They absolutely are related to spinors. You need to understand spinors to understand twistors.
Here's a rundown:
Points on the light cone in Minkowski -> 2x2 Hermitian matrices with determinant 0 -> 2 component spinor (phase doesn't matter)
If we don't care about the overall scale of a spinor, that selects only directions on the lightcone, and mathematically this is the Riemann sphere CP(1).
So now Lorentz transformations SO(3,1) become conformal transformations on the Riemann sphere which can be represented as SL(2,C) on spinors.
Now twistors bump everything up a notch:
>Instead of SO(3,1) we have SO(4,2)
>The null directions in SO(4,2) are a compactification of Minkowski space just like the Riemann sphere is a compactification of the plane
>So SO(4,2) is the conformal group on compactified Minkowski
>Instead of SL(2,C) on spinors this is represented by SU(2,2) on a four component object called a twistor
It's all trivial really
