Why does she need to make her face "an indifferent mask" and be careful what she says in public?
Katniss, in The Hunger Games, knows that there will be people that are going to be out to get her, that will hate her, and will try to sabotage her at all costs. By making her face "an indifferent mask" in public, she is trying to insure that she will not offend, anger, or show any emotion in regard to what is happening.
Facial emotion is a cue as to how one is feeling on the inside, typically. Katniss does not want anyone knowing how she really feels. Readers also know how much she hates the Hunger Games and the Capitol. She does not want that to show either.
For the "be careful of what she says in public" part can also work towards the end of the book. When she brings out the berries to kill them both for the end of the Hunger Games, the Capitol interprets this as an act of rebellion, so she needs to mask her motives and say that she only did it so that she and Peeta were able to stay together.