Definition and properties of support
First, the definition is clear: support is exactly "the fraction of transactions that contain a particular subset of items." This is a data-mining term, not a statistics term.

$supp(A)$ is the fraction of transactions that contain item $A$. $supp(B)$ is the fraction of transactions that contain item $B$. $supp(A,B)$ is the fraction of transactions that contain the subset $\{A,B\}$.

Now, your question really seems to be more about how the authors expound on this, which does seem to be a little unclear. As a statistic, support can basically mean, "What is the probability that this subset of items is contained in a random transaction?" That is, you can consider that $supp(A)=\mathbf{P}(A)$.

Then you can see that 
$$supp(A)supp(B) = supp(A, B) \iff A \text{ and } B \text{ are uncorrelated}$$
comes from the definition of statistical independence,
$$\mathbf{P}(A\cap B) = \mathbf{P}(A)\mathbf{P}(B) \iff A \perp B.$$