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.