What does giving face to mean?

Chances are, the phrase you are looking for is giving a face to, not giving face to (although the latter could conceivably be a variant on giving head, a colloquialism for oral sex). To give a face to something, such as for example the environmentalist movement, would be to offer some person or other visible symbol by which we can think of or visualize it. So, Ralph Nader could be said to be the face of environmentalism.