Eutrophication is an excess of certain nutrients in a body of water which stimulates the growth of aquatic plant life. The increase of plant growth, mainly algae, leads to a depletion of dissolved oxygen. 
This depletion can be so drastic that it asphyxiates other life, such as animals and non-algal plant life. Many fish and invertebrates depend on certain plants for their habitats, so even if they are not directly asphyxiated by the algae blooms themselves, they could still die out if their natural habitats are destroyed by the algae blooms. An increase in algae due to nutrient spike also reduces water clarity and visibility. Many fish rely on water clarity to see prey or predators. Thus, unclear water puts those species at risk. For these reasons, eutrophication threatens the delicate balance that healthy ecosystems require. 
How does eating meat contribute to eutrophication? Meat production creates an immense amount of animal waste, most of which is wasted instead of being recycled as fertilizer. Meat production also requires the production of animal feed, which in turn requires massive amounts of fertilizer. Agricultural runoff, such as animal waste and fertilizer, gets dumped into rivers and other bodies of water. For instance, slaughterhouses and factory farms from the Midwest dump their agricultural runoff into the Mississippi River, which of course leads into the ocean. Then, the nutrients feed algal growth and phytoplankton, which multiply before dying. Bacteria on the seafloor consume the tiny dead organisms, using up lots of oxygen in the process. All of the agricultural waste from the Mississippi River, along with the runoff being put directly into the water, has created an uninhabitable “dead zone” in the Gulf of Mexico, where native species are literally being suffocated to death. 
To stop the spread of eutrophication, meat consumption must be slowed if not ceased entirely. The increase in eutrophication in recent years is directly linked to the runoff from animal waste and animal feed fertilizer. If we stopped eating meat, then there would be vastly less agricultural runoff and thus less eutrophication.