What does a mm of rain mean?
It's confusing to measure liquid in units of length instead of volume, isn't it? Here's how it works.

"One millimeter of rain" is actually one cubic millimeter per square millimeter. On average, over the area you're talking about, each square millimeter has received one cubic millimeter of rain. If you divide n mm3 by 1 mm2, you get -- n mm!

The field of Dimensional Analysis deals with questions like this.

If there was "1 mm of rain", how much rain did a square meter receive? 1000 mm * 1000 mm * 1 mm = 1 000 000 mm3 = 1 liter. Pour one liter of water into a square container one meter on a side, and it forms a layer one millimeter deep.

To calculate "how much rain fell on Hong Kong" (expressed as total volume), you'd take the area of Hong Kong, and multiply it by 1 mm. That's usually not what people want to know, though -- they want to know how wet things got, and the volume-per-area number is most convenient for that.