How to write the turing machine processing operations?
The state marks the position of the head (as well as the current state): for example, in $q_00011$, the head is to the left of $0011$, and in $xq_1011$, it is to the left of $011$.

The notation $\sigma \vdash \tau$ means that the configuration $\sigma$ leads, after one step of the Turing machine, to the configuration $\tau$.

As David Richerby mentions in his comment, all of this is explained in your textbook at some point.