The paper in the comment above gives some cool visualizations, but I thought I would mention some easy-to-understand computational interpretations, which can help give intuition (at least to me), even if they are not exactly physical in nature.
Shape Analysis in 3D
See Zeng et al, Ricci Flow for 3D Shape Analysis.
Basically, the Ricci flow deforms an arbitrary 3d shape into one of $\mathbb{S}^2$, $\mathbb{H}^2$, or $\mathbb{R}^2$, depending on the shape's topology (and of course the resulting scalar Ricci curvature). The author's note that (in the 3D case) for a closed surface, if the total surface area is preserved during the flow, the Ricci flow converges to a metric with constant Gaussian curvature everywhere. Classic differential geometry is probably more intuitive.
Image Processing
See Appleboim et al, Ricci Curvature and Flow for Image Denoising and Super-Resolution. In it, the authors treat greyscale images as Riemannian manifolds and note that (with their particular construction of the image manifold) running the Ricci flow is essentially analogous to evolving (diffusing) the gradient of the image, rather than the pixel values.
Ito Diffusion Behaviour
The behaviour of Brownian motion on manifolds is such that negative scalar curvature is known to accelerate diffusion outward from the starting point (while positive slows it down). See e.g. Debbasch et al, Diffusion Processes on Manifolds. Maybe with this in mind, one can imagine Ricci flow to be "smoothing out" the behaviour of a diffusing particle on the manifold, in a way.
While it's true that $g_t$ has the same isometry group for any $t\in[0,\infty)$, it may not be the case that the normalized limit $\lim_{t\to\infty}g_t$ has the same isometry group. As a simpler analogy, consider the family of functions $f_t(x)=e^{-t}f_0(x)$. $f_t$ may not be translation invariant for any $t$, but $\lim_{t\to\infty}f_t$ is. The previous statements doesn't allow anything to be said about the isometries of this limit since isometry groups do not depend "continuously" on the metric.
What the discussion of isometry groups essentially says is that the "features" of an "ugly" space do not disappear in finite time, but the convergence results say that these features do get "arbitrarily small" in some suitable sense. In order to get rid of the features entirely, thereby introducing new symmetries, a limit is required.
Best Answer
The correct interpretation is indeed that $\def\isom{\operatorname{Isom}}\isom(g_0)\subseteq \isom(g_t)$ for all $t,$ i.e. that if $\phi$ is an isometry of $g_0$ then it is an isometry of every $g_t.$ This is a simple consequence of the isometry-invariance of the Ricci tensor: since $$\def\rc{\operatorname{Ric}}\rc_{\phi^*g} = \phi^* \rc_g$$ for any isometry $\phi$, the defining equation $\partial_t g_t = -2 \rc_{g_t}$ implies $$\partial_t (\phi^* g_t) = \phi^*(\partial_t g_t)=\phi^*(-2\rc_{g_t}) = -2\rc_{\phi^*g_t};$$ i.e. $\phi^* g_t$ is also a solution of the Ricci flow. Since $\phi$ is an isometry of $g_0$, these two Ricci flows have the same initial condition $\phi^*g_0 = g_0,$ so by uniqueness they must be equal for all time, i.e. $\phi^* g_t = g_t.$