"Way down here" probably refers to the belly. Westerners, especially Americans, will talk about a 'gut feeling', something we feel very deeply within us (as opposed to our hearts). The 'nice guys' were assuring everybody that they have deep feelings - feelings they experience at a deep physical level - for the common or little folks.

The expression "way down here" means feeling something deep inside ... By using quotes, the writer is suggesting that they're taking that expression from somewhere else, right? I would say the expression "way down here" comes from history, either a book or a movie, where people were more caring about others, and felt things "deep inside" In any case, the important detail is that this expression is in quotes -- which could be for emphasis (adding stress to the part of the sentence she would stress if she was speaking it out loud), or to show that she is borrowing that expression from another context...