An idiom is an expression that is not understood simply by the individual words. In this case, "gets my goat" to make someone angry or annoyed. Other examples of idioms include "it's raining cats and dogs" and "you've hit the nail on the head."
The children would most likely choose unhealthy options like fast food or junk food because they want sugar and don't know the benefits of healthy food. The family's provisions would likely be less healthy for this reason.
There are a few ways this could be answered, but judging by the fact that the question asks what people "do to" his statements, it makes sense to say they would discredit them.
