What is the best way to interpret, explain or somehow visualize the basic idea behind formal definition of Ricci Flow?
I am familiar with the hackneyed expressions like
"Ricci Flow is a non-linear analogue for the heat equation which smoothens metric",
or
"Ricci Flow describes the deformation of the Riemannian metric tensor on manifold".
However, I was looking for something similar to the balloon-under-pressure interpretation of the mean curvature flow, in particular of the surface tension flow:
Ultimately, I am looking for something that would make Ricci Flow concept clear for undergraduate students without diving too deep into technicalities of Ricci tensor and volume forms.
I'm late to the question, but this writeup by Terry Tao is the best intuitive, high level source on the subject that I know. I'll summarize it in my own words.
First we need to describe the notion of a Riemannian metric, since this is what is evolving under the Ricci flow. The Riemannian metric $g$ is a way to describe the shape of a manifold, by defining the lengths, areas and angles of that manifold locally. For example, standing on the surface of the earth, you only have a perspective of distances, angles and area with respect to that surface rather than the larger space $\mathbb{R}^3$.
The Ricci curvature of a point $x \in M$ tells us how "non-Euclidean" a Riemannian manifold is at $x$. Specifically, we can measure this by considering the area of a sector of a circle having angle $\theta$ and radius $r$. In Euclidean space, this area is $\frac{1}{2}\theta r^2$ and in a Riemannian manifold $M$, we can denote it by $|A(x,r,\theta,v)|$. The Ricci curvature of a surface is the (scaled) difference between these: $$ \text{Ric}(x)(v,v) = \lim_{r\rightarrow 0} \lim_{\theta \rightarrow 0} \frac{\frac{1}{2} \theta r^2 - |A(x,r,\theta,v)| }{\theta r^4/24}.$$
This is the formula for a 2-dimensional manifold (surface). More generally, the Ricci curvature measures the difference between the Euclidean volume and the manifold volume.
(Here is an image of how the area of a triangle differs in spherical, Euclidean, and hyperbolic space.)
The Ricci flow an intrinsic flow, meaning that it describes the evolution of a manifold using only its Riemannian metric (recalling that this is a local description of the shape at every point). It decreases the Ricci curvature in time. This means that a surface becomes "more Euclidean" at every point as the Ricci flow evolves it. Specifically, the Ricci flow is $$\frac{dg}{dt} = -2\text{Ric}$$ meaning that the Riemannian metric itself evolves, at a rate proportional to its negative Ricci curvature. This resembles mean curvature flow, in which a manifold evolves so that its mean curvature at every point "smooths out". The difference is that the Ricci is an intrinsic flow, measured by the local metric, as opposed to MCF which is extrinsic, where curvature is measured from the ambient Euclidean space.
As for your balloon-under-pressure interpretation I'm not sure. This probably has to do with the volume measured under the Riemannian metric. You might want to check out the surface tension viewpoint of minimal surfaces (surfaces that are stationary under MCF).