Intuitively what is meant by a left invariant vector field on a manifold?
To talk about left invariance, you probably want to assume your manifold is a Lie group, so that the vector field is left invariant under the (derivative of) the group action. Intuitively, this means that the vector field is entirely determined by the vector at the unit element of the Lie group. Given any other element of the Lie group, say g, the vector at g has to be (lg)∗(X0), where (lg)∗ is the derivative of left-multiplication by g and X0 is the vector at the unit element. So the Lie group action allows you to take a single vector and distribute it out over the manifold in a smooth, nonvanishing way.
The simplest example is Euclidean space Rn regarded as an abelian Lie group under addition. In this case, a left invariant vector field is simply a constant vector field in the usual calculus sense. All the vectors point in the same direction. The map (lg)∗ identifies the tangent spaces at each point in the usual “parallel transport” way.