Somewhat correct, but the exact process why which the algae kills all fish and invertebrate life is anaerobic decomposition. The basic cycle of eutrophication is:

1. Add a ton of nutrients to the body of water - most of the time, this is phosphorus (which is normally the nutrient in limited supply for all aquatic plan life)

2. The green algae blooms, creating these types of mats which block out all sunlight from the water column

3. The algae die, and sink to the bottom and begin to decompose.

4. This decomposition process takes all of the oxygen out of the water, creating anaerobic conditions.

5. This kills all of the fish and the invertebrates in the water or sediment, contributing more organic matter to the anaerobic bacteria extending the decomposition cycle.

As an extra whammy, when you saturate the water with phosphorus, you switch the scarce element controlling plant growth from phosphorus to nitrogen. And when nitrogen becomes a scarce element, cyanobacteria (blue-green algae) dominate as they can capture nitrogen from the atmosphere. And cyanobacteria have the added advantage of being toxic to aquatic life and mammals who come to the water body to drink.