Let $S$ be a regular surface and $p \in S$. Then there exists a neighborhood $W$ of $p$ in $S$ such that $W$ is the graph of a differentiable function which has one of the following three forms: $z=f(x,y)$, $y=g(x,z)$, $x=h(y,z)$
So basically what the lemma is saying is that there exists $W$ neighborhood of $p$ such that $W$ is the graph of a two variable function. Why is this result useful? Is there a more intuitive reformulation of this lemma?
You'll find about a dozen questions dealing with related matters on here. This criterion (more generally, for submanifolds of $\Bbb R^n$ of any dimension) is very useful for showing a subset is not a regular surface. For example, $x^2+y^2-z^2=0$ fails the "implicit" test at the origin, but that doesn't guarantee that it cannot be a regular surface. However, you can check that in a neighborhood of the origin, this surface fails the "vertical line" test in all three cases, and so cannot be the graph of any function in a neighborhood of the origin.
If you're particularly interested in the differentiability, try $x^2+y^2-z^3=0$. This fails to be a function at all if we solve for $x$ or $y$, but $z=f(x,y)=(x^2+y^2)^{1/3}$ is the graph of a function; however, that function fails to be differentiable at the origin, and so this, too, fails to be a regular surface.