Rain brings all of the airborne pollution that is in the atmosphere right into the pool. All the pollen, mold spores, seeds, all the stuff that comes from living things like trees, grass, weeds, flowers, etc. In addition to the nature stuff is man-made pollution - stuff like nitrogen compounds, sulfur compounds, ozone, CO2, CO, hydrocarbons, etc. The nitrogen and sulfur compounds (mostly nitrous oxide and sulfur dioxide - NOx and SO2) are fertilizer for any existing algae and food for any new pollen or spores brought in by the rain. In addition to being food, NOx and SO2 use up big amounts of chlorine. They eventually become nitrites and nitrates and sulfates in the water. CO and CO2 are used by plants during photosynthesis. Hydrocarbons are tough to destroy and chlorine wears itself out trying to oxidize them.

You will always have an algae bloom or new algae growing after a rain. It is best to superchlorinate as soon as you can after a rain.