heavier tails means that it is less sensitive to outlying data for logistic and probit This is the description of logistic regression and probit regression as generalized linear model from wiki:

Both the logistic and normal distributions are symmetric with a basic unimodal, "bell curve" shape. The only difference is that the logistic distribution has somewhat heavier tails, which means that it is less sensitive to outlying data (and hence somewhat more robust to model mis-specifications or erroneous data).

I don't understand the conclusion somewhat heavier tails, which means that it is less sensitive to outlying data. Heavier tails means the relatively high probability of extreme values (outliers). Why do we say it is less sensitive to the extreme values?

The statement is not about generating extreme values from either a logistic or a normal, it's about trying to fit a logistic or a Normal to pre-existing data that may have extreme values.
You have data, and you are trying to fit a model to it. However, a data point is not an outlier if it is "expected", in some sense, by the model.  Any extreme values in the data are more likely when viewed through the lens of a logistic distribution than when viewed through the lens of a Normal distribution, therefore, the parameter estimates don't get moved around as much to try to "explain" (fit) them.  This can be restated as the parameter estimates being less sensitive to outliers when we use a logistic distribution than when we use a Normal distribution.
