When working with maximum likelihood, we derive the estimated coefficients by taking the partial derivatives of it and equating them to zero. We then find the expression for each coefficient in terms of the data. These derivatives are called scoring equations, and robustlmm works by making these equations more robust. This package uses several tricks to ensure that the impact of abnormal observations on these equations is small.
A detailed explanation of the algorithm can be found at https://cran.r-project.org/web/packages/robustlmm/vignettes/rlmer.pdf.