In this saying, "rain" stands for unhappiness or unpleasantness. However, note that it is "a little rain," not "a lot of rain," so the implication is that there will be more sunshine (happiness) than rain.
"Must" here means that according to the laws of how the universe works, something will be sure to happen.
So, it means that some unhappiness, for certain, will come into everybody's life. It means that one has to expect a little unhappiness, at least. Life doesn't consist of sunshine every day, all day; there has to be some rain, too.