Spin $1/2$ motivation for twistor theory in general relativity?

Most standard GR textbooks approach the problem of curved space-time using the notion of tensor fields. Spin 1/2 particles, on the other hand, are defined using spinors which are semi-tensor like quantities. Although one can define these particles using the Dirac equation, it works in flat space-time... and there doesn't seem to be an intuitive and natural way to extend this equation to a general curved space-time. The issue here is that we are trying to define the transformation of spinor like quantities using tensor fields, which is quite artificial (at least how I interpreted Dr Woit's statement).

So a better way is to start with 2-component spinor fields. To ensure that a manifold M admits spin bundle, it should satisfy two conditions: (1) M must be space-time orientable (2)The topological invariant - second Stieffel-Whitney class must vanish

These Spinors are complex quantities and it can be shown that the real tensor fields are a special subclass of these spinorial quantities. So all of the standard tensor transformation that we use in GR can be derived from a general class of spinor transformation. In other words, given that M admits spin bundle, these 2 component spinor fields can be interpreted as more fundamental than tensor quantities. It follows that you can describe any arbitrary spinor field ( spin n/2) in a general curved space-time using a 2-component spinor treatment. (Point to note: this description is very particular to (3+1) dim space-time) Read: " Spinors and Spacetime " Volume I - R. Penrose, W. Rindler.

Twistors are defined using 2-components spinor fields only. A Twistor is essentially a pair of spinors that describes the momentum and angular momentum of a massless particle. In Twistor theory, you can describe massless spinor fields in a flat space-time using Twistor functions (via what is known as Penrose transform)...and this is a 1-1 correspondence. The motivation for Twistor theory is to give an alternate geometrization of conventional physics where your space-time plays a secondary role...and various fields and curvatures are essentially an emergent property. Either way, these 2-component spinor fields play a key role in all of this description. ( Suggested reading: "Twistor theory- An approach to quantisation of fields and space-time" by R. Penrose, MacCallum (1972) )