Understanding conclusions that functional regions are under negative selection?
Lets assume for simplicity that DNA is globally subjected to the same mutation rate (which is probably not a fully correct assumption).

Now let take a DNA region which is functional (what you meant by giving a fitness benefit), mutations in this region will occur as anywhere else in the genome. Some mutations will be deleterious and reduce the fitness of (or kill) the cell therefore, on the long run, cells that do not carry those deleterious mutations will propagate. This means that now if you look at the genome in the cell that propagated, certain genomic regions will contain less mutations than other because of the negative selection. They were "protected" against deleterious mutation.

Similarly, In multicellular organisms any mutations that reduces the fitness of the germline cells will be negatively selected (again functional regions will be more conserved).

The author was likely referring to regulatory regions. These are regions that do not encode proteins but are very important for gene regulations. Regulatory regions include enhancers, silencers, insulators and locus control regions (Maston GA, 2006). These regions will also be more conserved as deleterious mutations in regulatory elements could reduce the fitness of the cell/germline.