What does media savvy mean?

Being "media savvy" for a "personality or entertainer" refers to your familiarity with how the "media" operates.

Being media savvy would imply that you are familiar with the importance of such things as public image, press releases, publicity agents, good p.r.

A politician would be considered media savvy if he was adept at getting stories ran to support his causes, creating negative press on his political adversaries, getting stories "buried" or "killed" by means of persuasion, friendly & equitable or not.

In short it implies an understanding of how to best utilize the "media" to one's own advantage, or to the dis-advantage of an adversary or competitor.