Q:

The exact meaning of "for that matter"

What is the exact meaning of, "for that matter," in the sentence: 

He did not speak to her, or anyone for that matter?  

A:

It's an idiom you append to denote your generalization includes the prior case - something you use instead of "either"/"too", when the group of the other part of the sentence swallows the first one instead of supplementing it.

He didn't eat the cucumbers, or any food at all for that matter.
There are no teen male latinos in the sample group. There are no males in it at all, for that matter.
Yes, the hiking group has a first aid kit. It even has a fully supplied military field medic backpack, with a full field surgery kit, for that matter.
I didn't complain about the food. I have nothing but praise for the staff for their service, for that matter.

If we were to expand the idiom into a full sentence, it would be something like this:

He did not speak to her. And if we're talking about the matter of him speaking: he did not speak to anyone at all.

