To stiffen your back means to align your spine - in most cases it means you stand straighter and taller. A person may do this to try to focus or show courage. Looking at several examples, the person is often about to start a challenge or be hostile. We also say "his/her back stiffened", which is a reaction to an unpleasant situation, or bad news.

Do you have the full context? The person may also be preparing to lift something heavy. So it depends.