The adjusted response function isn't a diagnostic plot like an added variable plot, where you try to investigate or isolate the effect of a single predictor or term. Instead, it's a summary plot intended to give an idea of the response as a function of one predictor, averaged over the others.
Suppose you have predictors X1, X2, and X3. Suppose the fitted model is a function of those predictors:
Now suppose you want to look at this as a function of X2. You have N data points all together. Then you could compute
f(x) = average over i=1:N of f(X1(i), x, X3(i))
So you are computing the fitted values at X2=x, averaged over all of the sets of observed values of the other predictors X1 and X3. Notice that the function f() could include main effects like X1, squared terms like X1^2, and interaction effects like X2*X3. That's different from an added variable plot where the term is the focus. Here it's the predictor variable, which could contribute to multiple terms.
This was proposed by Bill DuMouchel on page 127 of "Computer Science and Statistics: Proceedings of the 20th Symposium on the Interface" in 1988.