I think there are two parts to your question. First, what is meant here by the term stabilize, and second does chaotic motion stabilize elliptical galaxies (and if so how)?
Meaning of 'stabilize': As long as they are not strongly interacting, galaxies are (to very good approximation) in virial equilibrium, which implies (virial theorem for self-gravitating systems)
$$
\langle v^2\rangle = GM/R,
$$
where $\langle v^2\rangle$ is the overall mean-square velocity, $M$ the total mass, and $R$ a typical radius. Thus, the stars (and other objects including dark-matter particles) must move by just the right amount. If virial equilibrium is not satisfied, the galaxy either collapses (if $\langle v^2\rangle < GM/R$) or expands/explodes (if $\langle v^2\rangle > GM/R$), i.e. is unstable.
The overall mean-square velocity $\langle v^2\rangle$ is the density-weighted mean of the local mean-square velocity $\overline{v^2}$:
$$
M \langle v^2\rangle = \int\mathrm{d}^3x\,\rho\,\overline{v^2}.
$$
Moreover, we can always express the (local) mean-square velocity as
$$
\overline{v^2} = \overline{\vec{v}}^2 + \sigma^2,
$$
the sum of the square of the mean velocity $\overline{\vec{v}}$ (accounting for ordered motions) and the velocity dispersion
$$
\sigma^2 = \overline{\left(\vec{v}-\overline{\vec{v}}\right)^2}
$$
(accounting for un-ordered or random motions).
Stability by random motions? There are various ways in which stars can move to achieve virial equilibrium. One possibility, realized in disc galaxies, is to move on co-planar near-circular orbits (with the same sense of rotation). In this case $\sigma\ll|\overline{\vec{v}}|\approx v_{\mathrm{rot}}$, and hence $\overline{v^2}\approx v_{\mathrm{rot}}^2$: the galaxy is stabilized by rotation.
Another option is to move on non-circular (elliptic or rosette-shaped) orbits with random orbital planes and senses of rotation. In this case
$\overline{\vec{v}}=0$ such that $\langle\sigma^2\rangle=GM/R$, i.e. the galaxies is stabilized by random motions. For elliptical galaxies $|\overline{\vec{v}}|\ll\sigma$, and hence $\langle v^2\rangle\approx\langle\sigma^2\rangle$: they are largely stabilized by random motions (even if some rotate).
Stability by chaotic motions? The term random motion refers to unordered motion, i.e. the deviation of individual velocities $\vec{v}$ from the (local) average velocity $\overline{\vec{v}}$. However, the term chaotic motion means something completely different, namely motion on chaotic orbits. The vast majority of orbits occupied in galaxies are regular and not chaotic (if the majority of orbits were chaotic, they could not support an aspherical shape, whilst spherical galaxies contain no chaotic orbits). Hence, the statement that elliptical galaxies are supported by chaotic motion is WRONG.
