If I understand you correctly, you are essentially asking why robust standard errors under clustering design are greater than those computed by the OLS? If that is indeed what you are asking, then typically OLS standard errors are calculated based on the assumption that Simple Random Sampling (SRS) was used. In practice, it is very common that clustered standard errors are greater than those produced by OLS under SRS. In contrast, if stratification was used (rather than clustering), standard errors tend to be smaller compared to the OLS.
If I understand you correctly, you are essentially asking why robust standard errors under clustering design are greater than those computed by the OLS? If that is indeed what you are asking, then typically OLS standard errors are calculated based on the assumption that Simple Random Sampling (SRS) was used. In practice, it is very common that clustered standard errors are greater than those produced by OLS under SRS. In contrast, if stratification was used (rather than clustering), standard errors tend to be smaller compared to the OLS.
This Chapter says the following: "The greater the correlation among units within a group (that is, the bigger Intraclass correlation is) the greater the impact on the standard error." Please see page 12/20 of this document for more details.