Regression is a way to model the relationship between the conditional expected value and the predictors. But in the robust regression we don't have the expected value (say, arithmetic mean for Gaussian response), but rather an M estimator - truncated mean, winsorized mean. So, when it comes to interpret the coefficients in a linear regression, it's simple: "a unit change in X1, holding constant all others, causes, in AVERAGE, beta change in the response". Now, how to replace the "AVERAGE" in robust regression?
With robust regression you are still estimating the conditional mean function $E(Y|X) $ so that the interpretation that you suggest carries over. The difference is that robust estimators require fewer assumptions on the error and the covariates than the typical least squares estimator.