I have often heard that the theory of Twistor spaces is ``a complex analogue" of the Radon transform. What is the precise connection ?

The correspondence between the Radon transform (from a space of real-valued functions on $\mathbb{R}^2$ to the space of functions on the manifold of straight lines in $\mathbb{R}^2$) and the Penrose transform (from $\mathbb{CP}_2$ to the dual projective space $\mathbb{CP}^\ast_2$) is easiest to work out when lines in the Radon transform are replaced by great circles on a sphere. This special spherical case of the Radon transform is called the Funk transform, and it corresponds to the Penrose transform when $\mathbb{CP}_2$ is restricted to $\mathbb{RP}_2$, see The Funk transform as a Penrose transform (1999).

To understand the relationship between the Penrose and Radon transforms, it's hard to do better than the outline given by Atiyah in [1]. See chapter VI, section 5 (pages 78--81).

(It even looks like there's a PDF available online if you search.)

[1] Atiyah, M. "Geometry of Yang-Mills Fields", Annali della Scuola Normale Superiore di Pisa (1979).