Intuition behind eigenvalues of an adjacency matrix
(Disclaimer: This answer is about eigenvalues of graphs in general, not the second eigenvalue in particular. I hope it is helpful nevertheless.)

An interesting way of thinking about the eigenvalues of a graph $G = (V, E)$ is by taking the vector space $\mathbb{R}^n$ where $n = |V|$ and identifying each vector with a function $f\colon V \to \mathbb{R}$ (i.e., a vertex labeling). An eigenvector of the adjacency matrix, then, is an element of $f \in \mathbb{R}^n$ such that there is $\lambda \in \mathbb{R}$ (i.e., an eigenvalue) with $A f = \lambda f$, $A$ being the adjacency matrix of $G$. Note that $A f$ is the vector associated with the map which sends every vertex $v \in V$ to $\sum_{u \in N(v)} f(u)$, $N(v)$ being the set of neighbors (i.e., vertices adjacent to) $u$. Hence, in this setting, the eigenvector property of $f$ corresponds to the property that summing over the function values (under $f$) of the neighbors of a vertex yields the same result as multiplying the function value of the vertex with the constant $\lambda$.