What does beta distribution "has support" mean? This post says

the beta prior has support over all valid probabilities and only over
  valid probabilities

About Support, wiki says

In mathematics, the support of a real-valued function f is the subset
  of the domain containing those elements which are not mapped to zero.
  If the domain of f is a topological space, the support of f is instead
  defined as the smallest closed set containing all points not mapped to
  zero. This concept is used very widely in mathematical analysis.

What does beta distribution "has support" mean here? Can anyone give an example to illustrate this? Such as this one
 A: For a single probability parameter, the interval $\mathscr{P} \equiv [0,1]$ is the set of "all valid probabilities and only valid probabilities".  Thus, when they say that the Beta distribution "has support over" $\mathscr{P}$, what they mean is that for any random variable $p \sim \text{Beta}$ we have:
$$\begin{equation} \begin{aligned}
\mathbb{P}(p \in \mathscr{P}) &= 1, \\[6pt]
\mathbb{P}(p \in \mathscr{S}) &<1 \text{ for every closed (measureable) set } \mathscr{S} \subset \mathscr{P}.
\end{aligned} \end{equation}$$
In other words, the interval $\mathscr{P}$ is the smallest closed set containing $p$ with probability one.  Informally, this can be thought of as the closure of the set of possible values of $p$.  (Note that the same reasoning occurs for the Dirichlet distribution when you extend to a probability vector.)  The support can also be thought of as the closure of the set of values with non-zero density, so another way to think of the support here is that:
$$\mathscr{P} = \text{cl} \{ p \in \mathbb{R} | \text{Beta}(p|\alpha,\beta) > 0 \}.$$
A: I think it is better read this way:

the beta prior has and only has support over all valid probabilities

However, I think it is a bit redundant in terms of its meaning? 
A: I wanted to provide a non-technical answer to your question.  
The beta distribution is commonly used in the context of modeling proportions. 
As an example, let’s say you select (at random) 100 geographic sites for a study and you keep track for each site what proportion of the site’s area can be classified as “forest”.  (If 20% of the area of the first site can be classified as “forest”, the proportion of interest is 0.20, etc.)
If you denote by X the random variable defined as “proportion of site’s area that can be classified as a forest”, then you might want to assume that X follows a beta distribution.
What does it mean that this beta distribution has a support of (0,1)?
It simply means that the possible values of X span the interval (0,1). In particular, the 100 realized values of X collected in your study would all be expected to be strictly greater than 0 and strictly less than 1.
This is fine if you selected your sites so as to always include a mixture of forest and grasses. 
But what if you’re interested in sites which might be all forest, all grasses or a combination of both? 
Then you might have to model X via a zero-and-one inflated beta distribution, whose support is the interval [0,1]. In that case, the support of the distribution of X tells you that the possible values of X could live anywhere inside the interval [0,1] (including at the edges of this interval). For the example of with 100 sites, this might mean that a fraction of those sites would have no forest on them, a fraction would have some forest and a fraction would have only forest on them.
