If there are no obstructions to deformation, i.e. $h^1=0$, then Kodaira's deformation theorem says that the map admits a family of deformations, forming a manifold, near that given map, with tangent space given by $H^1$$H^0$. You think of genus zero curves as like lines, or line segments, and deforming a line while staying in the variety should be like trying to move around on a set of points in Euclidean space while still being able to move your line segment with you, something you can do when points of the set are connected by line segments, i.e. in a convex set.
For example, if $f$ is the identity map of the projective line, then the space of deformations of $f$ is the space of maps of degree 1, i.e. the space of projective linear transformations of the projective line. The tangent space at the identity element of the group of projective linear transformations is the Lie algebra of vector fields on the projective line, i.e. $H^0$ is the space of linear vector fields, i.e. the space of $2 \times 2$ traceless matrices, so 3 dimensional.
Another example: if $f$ is constant, then the deformations are constant maps, and if the target $X$ is smooth at the image point $x=f(\mathbb{P}^1)$, then $H^0=T_x X$.
Kodaira's book on deformation theory is where I learned about this stuff. Maybe there are better references, especially in the algebraic category.