What does the minimal eigenvalue of a graph say about the graph's connectivity?

The smallest (in absolute value) eigenvalue $\lambda_1$ of the Laplacian (I don't know if this agrees with your definition of $\lambda_1$; I recall not liking Chung's definitions) measures how fast heat dissipates on the graph.

More precisely, for a (finite, for simplicity) graph $G = (V, E)$ with Laplacian $\Delta$ (regarded as an operator acting on functions $V \to \mathbb{R}$), the heat equation for a function $u(v, t) : V \times \mathbb{R} \to \mathbb{R}$ $$\Delta u = \frac{\partial u}{\partial t}$$

has general solution $$u(v, t) = \sum_{k=0}^{n-1} e^{\lambda_i t} f_i(v)$$

where $f_i : V \to \mathbb{R}$ is the eigenvector of $\Delta$ with eigenvalue $\lambda_i$ (and recall that $\lambda_0 = 0$). As a function of $t$, the dominant term above is the term $e^{\lambda_1 t} f_1(v)$, which controls the decay of the heat distribution $u$ as $t \to \infty$. Intuitively you should think of the heat equation as describing a "continuous-time random walk" on the graph. If such a random walk mixes quickly (so $\lambda_1$ is large in absolute value), then the graph is highly connected; if not, then perhaps the graph has bottlenecks of some kind. Cheeger's inequality makes this intuition precise.