In this context, the fact that the space is homogeneous means what follows.
Assume to have an isolated mechanical system. If I fix an initial condition (position and velocity of every point of the physical system at the initial instant of time), I find a corresponding evolution solving the equation of motion. What happens to the resulting motion of the system if I change that initial condition (at the same initial time) by means of
given translation $T$?
In inertial reference frames the answer is simple even if by no means physically obvious:
The resulting motion is exactly the previous one, with the only difference that at each instant of time every reached configuration has to be changed by the action of that initial $T$.
If it happens for every mechanical system and every translation, the rest space of the used reference system is said to be homogeneous. This is the case for inertial frames, but this property generally fails for non-inertial reference frames.
Replacing translation for rotation, the notion of isotropic space arises. Replacing rotation with time translation, one has the notion of homogeneous time.
There is a remaining notion due to pure Galilean invariance, i.e.
$t\to t$, ${\bf x} \to {\bf x} + t {\bf v}$ for all possible vectors ${\bf v}$.
Dynamics in inertial frames is also invariant under that class of transformations.
(All that is related with the 10 one-parameter subgroups of Galileo's Lie group).
Coming to the Lagrangian formulation, the Lagrangian of a point can, in principle, be a function of time, position and velocity of the particle.
It is not so difficult to see that, in view of the structure of Eulero-Lagrange equations, imposing (1) homogeneity, (2) isotropy, (3) time homogeneity and (4) invariance under pure Galilean transformations for a point of matter, its Lagrangian has to assume the standard form $k \dot{x}^2$. (I am not sure whether or not the invariance under pure Galilean transformations can be dropped as an hypotheses.)