To take it literally, to start a fire means to set fire to something in such a way that it keeps burning. When people are camping, indeed, if they intend to have a campfire for warmth or cooking, they will need to start a fire. You might also use the term light a fire. However, if there's a large, uncontrolled fire, start will be used rather than light. If it's a managed, deliberate, contained fire, I believe that both are acceptable, and which is more common depends on dialect. I would tend to say light when it's a controlled fire, except possibly in the case of camping. And I don't go camping.
It is also used metaphorically, the most well-known example possibly being Billy Joel's song We Didn't Start The Fire. Where any idea or phenomenon is perceived as hard to control and especially if it is seen as destructive, then the person or people who gave rise to it may be said to have "started the fire" in a metaphorical sense.