The AMOC relies on a cycling of warmer water in the tropics and cooler water in the Arctic Circle to generate the circulatory current. The warm water flows north, cools and sinks below the thermocline, then flows back to the tropics. It is not a loop on the surface like tidal flows, but rather a loop in an elongated cross sectional view that stretches right around the Indian and Southern Oceans as well as the Atlantic and is, in effect, a gigantic natural heat pump moving energy from the tropics to the North Atlantic ocean. The basic idea behind the potential shutdown of the flow is that as the temperature differential declines, so does the energy in the system, resulting a slowdown of the current and, ultimately (if taken to a logical conclusion), it stopping altogether - just as a heat pump would once the temperatures on either side have the pump have equalised.
In terms of impact, there's a bit more to it than that to do with variations in salinity between different parts of the ocean, which in turn being compounded with the inflow of fresh water from the melting Arctic ice cap and (mostly) Greenland's glaciers, that it also bring nutrients essential for the supporting the marine life in the Atlantic, plays a key part in sequestering the vast amounts of CO2 the ocean captures into the deep ocean (which is a whole other feedback loop). Even if it doesn't stop altogether, but only slows significantly, the impact on the entire biosphere, and especially around the North Atlantic, is going to be profound.