The cluster term seems to have behaved as intended: it prevented you from making an unwarranted premature claim of significance.
Standard statistical tests (like your Likelihood ratio, Wald, and Score tests) assume independence among events. When there can be recurrent events within the same individual, the events experienced by that individual can't necessarily be thought of as independent. The cluster term leading to the robust estimate of variance takes that into account.
The "significance" of the other tests only represents what would have been the case if each event depended independently on the covariate values in place at the event times, without regard to prior events experienced by an individual. The subsequent loss of significance in the robust model suggests that there are tendencies of some individuals to have more events than other individuals, in ways that are not accounted for by covariate values. Perhaps having one event makes it more (or less) likely to have a subsequent event even if covariate values are the same. Perhaps the covariates you have identified are actually responsible for those differences among individuals but there simply weren't enough individuals to document that. Or perhaps you simply haven't yet found the correct covariates that account for survival differences.
Without further information about your data and model it's hard to say more about what's going on. There could be some interesting phenomenon underlying your result, but working that out would take more detailed modeling of the recurrent event structure. You will have to balance the effort required for that work against the possibility, indicated by the non-significant robust result, that there is northing really there to find.
clusterterm was trying to handle, the numbers of cases and events, and the actual output from the summary of the model. Without that type of information it will be really hard to provide a helpful and reliable answer. In particular, all survival data are necessarily longitudinal, so unless there are things like repeated events aclusterterm might not be needed. Please put that information into the question itself by editing it, as information provided in comments is easy to miss and can end up lost. $\endgroup$ – EdM Sep 8 '20 at 18:40