This is closely related to the fact that in a Euclidean space, coordinate translations can be generated by performing two successive rotations around different points, as isotropy is essentially rotation invariance and homogeneity translation invariance. Suppose we have a rotation $R(\vec{r}_0)$ respect to $\vec{r}_0$ defined through the action on any point $\vec{r}$ as
$R(\vec{r}_0)\vec{r}=\vec{r}_0+R(\vec{r}-\vec{r}_0)$
Then clearly $R(\vec{r}_0)\vec{r}_0=\vec{r}_0$. For simplicity, we simply use $R$ to denote a rotation respect to the origin. Then for any point $\vec{r}$, two successive rotations around origin and $\vec{r}_0$ respectively would give
$R^{-1}(\vec{r}_0)R\vec{r}=\vec{r}_0+R^{-1}(R\vec{r}-\vec{r}_0)=\vec{r}+(I-R^{-1})\vec{r}_0$
Then for any translation $\vec{a}$, we can choose the coordinate system such that $\vec{a}=(a,0,0)$, then set
$\displaystyle R^{-1}=\left(\begin{array}0 &-1&\\1&&\\&&1\\\end{array}\right)$
and $\vec{r}=(a/2,a/2,0)$, we get
$\vec{r}+\vec{a}=\vec{r}+(I-R^{-1})\vec{r}_0=\vec{r}_0+R^{-1}(R\vec{r}-\vec{r}_0)=R^{-1}(\vec{r}_0)R\vec{r}$
Then if the space is invariant under rotations with respect to any point, it will be invariant under translation. In curved spacetimes, instead of global rotations, we need to consider Killing vectors. And similarly, existence of Killing vectors for isotropy at every point implies the existence of Killing vectors for homogenity. For details, see Chapter 13 of Weinberg's extraordinary book, Gravitation and Cosmology.