In a random-intercept model with between-group variance tau squared and within-group variance sigma squared, the intraclass correlation is tau squared divided by their sum. It answers two questions at once: how correlated are two rows from the same group, and what fraction of the outcome’s variance is explained by group membership.
Its practical bite comes through the design effect, 1 + (m - 1) times the ICC for clusters of size m, which converts nominal sample size into effective sample size. Even a small ICC destroys most of the information in large clusters, which is why ignoring grouping produces standard errors that are far too small, especially for predictors that vary at the group level.