The cluster starts with a randomly chosen particle and grows by adding particles with some probability (based on the Boltzmann factor). Particles that strongly interact with cluster are more likely to be part of the cluster. Then a normal trial move is made of the cluster as a whole.
Importantly, the potential and temperature chosen for this operation are not the same as the potential and temperature of the system. In fact, the cluster size can be controlled by adjusting the temperature (at infinite temperature each cluster is only a single particle). This cluster temperature is varied during the simulation run to sample moves with different cluster sizes.
See the paper for more details about controlling the range of energies involved in the cluster (and the cluster interface energies).