Cover and Thomas's book is indeed the right place to learn about this.
The statement basically follows by convexity, in the form of Jensen's inequality. Here is the way it is usually presented:
Let $f$ be the probability density of a real random variable. Then the Shannon entropy is given by
$-\int f\log f dx$
You want to prove among all real random variables with finite Shannon entropy and variance equal to $1$, the Shannon entropy is maximized only for Gaussians.
Given two probability densities $f$ and $g$, since $\log$ is a concave function, Jensen's inequality tells us that
$\int f \log (g/f) dx < \log \int f(g/f) dx = \log \int g dx = 0$
Moreover, since $\log$ is strictly concave, equality holds if and only if $g = f$. If you now set $g$ equal to the probability density of a Gaussian with the same variance as $f$ and plug in the explicit formula for $g$, you get what you want.
We can find some attempts for generalizing the notion of entropy of discrete random variables to random variables with general distribution function.
A straightforward way is to employ Riemann sum of the distribution function. So we start with a discrete random variable and then by making the intervals small enough, the entropy function is obtained. Denote the quantized random variable by $X_\delta$ where $\delta$ is the size of the intervals. If the probability density function $f$ is integrable, we can see for small $\delta$ (Cover-Thomas p. 248):
$$
H(X_\delta)\approx h(X)-\log\delta.
$$
By choosing $\delta$ equal to $2^{-n}$, i.e. $n$ bit quantization, we get
$$
H(X_\delta)\approx h(X)+n
$$
which represents how many bits we need to describe $X$ with $n$ bit accuracy.
This shows somehow the relation between differential entropy and discrete entropy. Note that when $\delta\to 0$, $H(X_\delta)\to\infty$.
Another point is that the mutual information does not change using this method, namely if $\delta\to 0$:
$$
I(X_\delta;Y_\delta)=I(X;Y).
$$
The generalization is attributed to different people, among them mainly Kolmogorov and Rényi:
A. N. Kolmogorov. On the Shannon theory of information transmission in
the case of continuous signals. IRE Trans. Inf. Theory, IT-2:102–108, Sept.
1956.
J. BALATONI and A. RENYI, Remarks on entropy (in Hungarian with English and
Russian summaries), Publications of the Mathematical Institute of the Hungarian
Academy of Sciences, I (1956), pp. 9--40.
Renyi introduced the following random variable ($[]$ is the integer part)
$$
X_n=\frac1{n}[nX].
$$
Note that this is nothing but looking at the intervals $[\frac kn,\frac{k+1}n)$.
Suppose that $H([X])$ exists, which is denoted by $H_0(X)$ in the original paper. The lower dimension of $X$ is defined as following
$$
\underline d(X)=\liminf_{n\to\infty}\frac{H([X])}{\log n}
$$
and upper dimension of $X$ as:
$$
\overline d(X)=\limsup_{n\to\infty}\frac{H([X])}{\log n}.
$$
Now if $\overline d(X)=\underline d(X)$, we simply talk about the information dimension of $X$, $d(X)$ and we define the following:
$$
H_{d(X)}(X)=\lim_{n\to\infty} (H(X_n)-d(X)\log n).
$$
Renyi proved that if $X$ has an absolutely continuous distribution with the density function $f(X)$ and finite $H([X])$, then we can say:
$$
d(X)=1\\
H_1(X)=h(X).
$$
This is what we discussed above for $\delta=\frac 1n$:
$$
H(X_n)=h(X)+\log n
$$
Kolmogorov instead introduced the notion of $\epsilon-$entropy which is defined for random variables in abstract metric spaces which is more general.
To answer your question, We can keep the same intuition as the discrete case for differential entropy at least when we use it for finding mutual information or KL-divergence.
For the entropy itself, we have to alter our intuition a little bit. The entropy of discrete random variable means the minimum bits we need to compress the random variable. But for random variables with uncountable supports, we can always "compress" it with another uncountable set of same cardinality (any one-to-one and onto mapping does that). But different random variables with uncountable supports can have different differential entropy.
Best Answer
Both the Gaussian maximum entropy distribution and the Gaussian solution of the diffusion equation (heat equation) follow from the central limit theorem, that the limiting distribution of the sum of i.i.d. random variables with given average and variance is a Gaussian. The connection between the central limit theorem and the diffusion equation, which describes a random walk with zero mean and unit variance, is obvious; the connection between the central limit theorem and the maximum entropy distribution is less obvious. For pointers to the literature on the latter connection, see these MO question and answers.