The two posts below explain very well the concept of RMSE (root mean square error) and also justify mathematically why it’s better than other similar residual based measures.
However, intuitionally to understand it, RMSE directly explains how close predictions are to the actual values as RMSE has the same unit as the response variable. A regression model with good R2 value or a good adj R2 value might still have an RMSE that’s unacceptable for the use case. This is when the model will have a low variance but high bias.