I try to explain how I understood that (that does not mean my understanding is correct!).

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Based on the way Penrose uses "transitive" to explain both the bundle of future timelike directions and also the homogeneity of space (preceding paragraph), I think what it means is "homogeneity of velocities" so that spacetime looks the same for every state of motion allowed by a vector in the interior of a future light cone. This is pretty fuzzy.

The "homogenety of space" means there is no special point (event) of Minkowski space. The same happens in an Euclidean space: there is no special point.

Then "homogeneity of velocity" means that there is no special velocity, i.e. that there is no special inertial frame, i.e. the special relativity principle. Every velocity can be obtained from any other one (transitivity) by an (active) Poincarè transformation, equivalent to move from one inertial frame to another.

The same happens in an Euclidean space: there is no a special direction at a given point, i.e. Euclidean space is isotropic.

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Presumably the base space of the bundle is the 3-dimensional space of allowed velocities. What, then, are the fibers - the 3-dimensional space orthogonal to the implied time direction ?

I think the "bundle of future timelike directions" is the (future timelike subspace of) tangent fiber bundle (chapter 15.5) of Minkowski space, which elements are the pairs (P,T) of an event P (t,x,y,z) (point of Minkowski space) and a tangent vector T at this point (i.e. a tangent to a world line, i.e. a 4-velocity vector). Hence the base space is the 4-dimensional Minkowski space itself, and a fiber is the space of 4-velocities allowed to a particle.

The 4-dimensional velocity space can be reduced to 3-dimensional (so making it very similar to classical velocity space) imposing that tangent vector length is unity, i.e. selecting an affine parameter (see chapter 14.5) for world lines: the affine parameter is the proper time for a particle being at that event and moving with that velocity. And then we can describe this unit vector with 3 parameters, that are the components of classical velocity.

This is the analogue of what we do in Euclidean space. We can describe a direction by a 3-dimensional vector; but its length is arbitrary, so we may decide to restrict ourselves to unit length vectors; and then we can describe this unit vector by 2 parameters, i.e. two direction angles.