Let $G = (V, E)$ be a finite graph and let $\Delta$ denote its Laplacian (here we take the convention that the Laplacian is positive semidefinite). For simplicity name the vertices $1, 2, ... n$. Then $\Delta$ is the matrix associated to the quadratic form
$$q(x_1, ... x_n) = \sum_{(i, j) \in E} (x_i - x_j)^2$$
which one might call the Dirichlet energy. Recall that, for an appropriate change of variables, such a quadratic form can be written as
$$q(z_1, ... z_n) = \sum_i \lambda_i z_i^2$$
where $\lambda_i$ are the eigenvalues of $\Delta$. Thus to determine the multiplicity of the zero eigenvalue it suffices to determine when $q$ can be zero. But from the first expression it should be clear that $q = 0$ if and only if $x_i = x_j$ whenever $(i, j) \in E$; that is, whenever $(x_1, ... x_n)$ determines a function
$$x : V \to \mathbb{R}$$
which is constant on each connected component of $G$. The dimension of this space of functions is clearly the number of connected components of $G$, and the conclusion follows.
I can make two sort of hand-wavy observations. From basic linear algebra properties we know that given a matrix $ \mathbf{L} $
\begin{equation}
\sum_{i}d_{i}= \sum_{i}\lambda_{i}
\end{equation}
where $ \lambda $ are the eigenvalues, and $ d $ are the the diagonal elements of your matrix $ \mathbf{L} $.
The elements on the diagonal of the Laplacian correspond to the degrees of the vertices. Assuming no isolated vertices and that the graph is simple (i.e weights are either $0$ or $1$), $ d_{i} \geq 1 $.
If most of your eigenvalues are equal to $1$, then their sum is also likely a small number. That means that the sum of degrees is also a small number, which means that your graph is probably sparsely connected.
The next observation has to do with star graphs. In general, complete bipartite graphs with $n$ and $m$ vertices on each group respectively, have Laplacian eigenvalues:
- $n$ with multiplicity $m-1$
- $m$ with multiplicity $n-1$
- $0$ with multiplicity $1$
- $m+n$ with multiplicity 1
Consider the star graph below which is a complete bipartite graph with $n=1$ and $m=8$
Its eigenvalues are $1$ with multiplicity $7$, $0$ with multiplicity $1$ and $9$ with multiplicity $1$.
So, if your graph contains starlike subgraphs, then your Laplacian is going to have a bunch of $1$-eigenvalues. I include two examples below (the random isolated component belongs to the graph on the right).
. 
Both of these graphs have eigenvalue $1$ with multiplicity $9$.
Best Answer
This question arises regularly. So far nobody has been to say anything useful about the multiplicity of zero in general. For special classes, there can be something. Thus for trees the multiplicity of zero is the number of vertices not covered by a matching of maximum size.