If you multiply both members of the equation (3.9) by $J$, you'll get:
$$
J\dot{q} = JJ^+v + \left( J -JJ^+J \right)\dot{q_0}.
$$

Then, we can exploit that $JJ^+=I$, obtaining:
$$
J\dot{q}=v,
$$
which is the well-known forward differential kinematics law.

Until now, we didn't specify anything about $\dot{q_0}$, which indeed can be whatever vector of the same size of $\dot{q}$. Indeed, we can plug whatever $\dot{q_0}$ into (3.9) that the equation will still hold.

Further, the equation (3.9) is clearly made up of two contributions:
- $J^+v$ that is the primary task.
- $\left( I -J^+J \right)\dot{q_0}$ that is the secondary task.

Because $\dot{q_0}$ can be anything, we say that the secondary task does not interfere with the primary task thanks to the projector $\left( I -J^+J \right)$.

Therefore, the flexibility given by the redundant mechanism at hand (i.e., dealing with the pseudoinverse $J^+$ and not the pure inverse $J^{-1}$ means that the manipulator is redundant) allows us to employ $\dot{q_0}$ for achieving a supplementary goal (i.e., the secondary task).

Usual secondary tasks are implemented to stay away from the joint bounds or to improve manipulability.

To this end, one can thus establish the following:
$$
\dot{q_0} = k_0 \frac{\partial w(q)}{\partial q},
$$
where $w(q)$ is some sort of function of the joints $q$ that we aim to minimize through gradient descent.

For example, if we want to stay away from the joint limits while converging to the primary target defined by $v$, we could do:
$$
w(q) = -\frac{1}{2n}\sum^n_{i=1}\left( \frac{q_i-\bar{q_i}}{q_{iM}-q_{im}}, \right)^2
$$
with $q_{iM}$, $q_{im}$ the upper and lower bounds of joint $i$, and $\bar{q_i} = \frac{q_{iM}+q_{im}}{2}$.