We use product rule when our thing consists of a sequence of the two things. For example if we need to visit city 1 and then city 2 and there are n roads to city 1 and then m roads from city 1 to city 2 then there are n*m ways of this trip as this trip is a sequence of two trips.

We use addition rule when our thing may be taken alternatively from the two sets and these sets do not intersect. For example, we need to get to city 1. We can do it by train (n ways) or by bus (m ways). Then there are n+m ways to make this trip.

Hope everything is clear now. Please ask me if something is not clear .