Interpreting probabilistic time turning machines
As the notation implies, the probability is taken over the choice of $r$. So it's the fraction of $r$s which the machine $A$ "accepts" (answer Yes). You're right that the random variable $A(r)$ has two possible values, Yes and No, and so for example the probability to get Yes and the probability to get No add up to $1$.

We usually think of $A$ as a statistical test. For example, $A$ might test the the number of $1$s in $r$ is roughly $\ell/2$, say between $\ell/2-\sqrt{\ell\log \ell}$ and $\ell/2+\sqrt{\ell\log \ell}$, and then it accepts with probability $1-2/\ell^2$ or higher (using Chernoff's inequality).

Something you are confused about is the branching. The machine $A$ can read its input bits many times. There is always one computation branch. After $\ell$ steps, there is still only one computation branch. On the other hand, at the outset there are $2^\ell$ different computations to consider.