It can be thought of as the total amount of rain that has fallen since the last fully dry day. Maybe yesterday was dry, in which case the storm rain would simply be today's rain. Or maybe it has rained every day for a month, in which case the storm total would be much greater.
A dry day is defined as a day with no more than 1 rainfall tip, since that could have potentially been caused eg by dew alone.
Calling it storm rain is a slightly unfortunate term IMHO. I imagine that it really relates to a mostly dry climate that is punctuated by heavy and potentially long-duration storms. But other climates might often have a little rain every day for a week, which personally I wouldn't call storm rain at all.