What does put out the fire while it's still small mean?

It means that if the fire is very little, it can be put out with little effort. The same can be true about any problem. Don't put off taking care of them or they will be a real problem later on.
For example, if a kid is thinking about trying weed, his parents need to sit down and talk to him and explain things (the small fire).
If they don't, he may end up dealing and using some heavy drugs (big fire) and the whole family will be in for a rough time.