Clusters are velocity dispersion-supported, not rotation-supported. Their net rotation is minimal. So when the velocity dispersion drops at larger radii, that really means that orbital velocities as a whole are going down with radius. That's expected for a system where the mass is concentrated toward the center (e.g. the Solar System).
So why does the velocity dispersion rise as a function of radius for $R\lesssim r_{200}$ in systems that are merging (left-hand panel), while it falls for systems that are not merging (right-hand panel)? A primary principle responsible for this effect is as follows.
For collisionless systems accreting in a cosmological context, material is approximately stratified, such that recently accreted material remains (on average) at higher radii than material that was accreted earlier. This is because material that accreted later has more energy and angular momentum than that accreted earlier. Consequently, the mass at each radius is connected to how rapidly the system was accreting at the time that its overall size was (something proportional to) that radius (e.g. Ludlow et al. 2013).
In particular, if the system accreted material rapidly at recent times, it will have more mass at higher radii.
Merging systems are biased toward having a high recent accretion rate. Hence, they tend to have more mass at higher radii. How this connects to orbital velocities can be understood on an approximate level by thinking about the velocity of a circular orbit at the radius $r$ in a spherical system,
$$v_\mathrm{circ}=\sqrt{GM(r)/r},$$
where $M(r)$ is the mass enclosed within the radius $r$ (not the total mass). $M(r)$ always increases with $r$. If the mass of the system is concentrated toward the center, $M(r)$ grows slowly with $r$, and potentially $v_\mathrm{circ}$ decreases with radius. If the system has a lot of mass at higher radii, however, then $v_\mathrm{circ}$ could potentially increase with radius (this happens specifically if $M(r)$ grows more quickly than $r$).
Conversely, non-merging systems are biased toward having a low accretion rate, so they tend to have less mass at higher radii. So for them, it makes sense that $v_\mathrm{circ}$ should decrease with radius. While the velocity dispersion is more complicated than $v_\mathrm{circ}$, it scales similarly.