It is crucial to understand that ‘Planet Earth’ is a closed system. Meaning that energy and emissions remain within, basically, the realm bounded by our atmosphere.
Heat is energy. If you add heat to the atmosphere, this stays in the system as increased amounts of energy and fuels more and more extreme weather events, such as heatwaves, torrential rain, floods, storms. We are seeing that all over the globe now: the crisis isn’t looming any longer. It is happening.
Emissions: We all know about CO2 by now. But as a greenhouse gas, methane is much more potent. Having been locked away in the ground in permafrost regions for millennia, it is now being released into the atmosphere as the soil thaws.
Melting ice has two massively dire consequences: if it is ice covering land - such as glaciers, or the ice sheets covering the Arctic, Antarctica and Greenland - it will eventually end up in the oceans and raise sea levels. A temperature rise of 3°C, which is what we are currently (late 2021) heading for by the end of the century, will put many coastal areas (think Bangladesh, Pacific atolls, Florida, Manhattan) under water. Not deep water, but still render the land uninhabitable. Across the globe, much low-lying land will get lost. And more elevated areas will face an influx of millions of coastal climate refugees.
The other thing melting ice does is create a feedback loop: white ice reflects sunlight (and thus heat) back into space; that’s called the albedo effect. If there’s no ice, the ground or, worse, the much darker ocean absorbs the sunlight, then gives the excess heat back to the atmosphere, which heats up some more - and then more ice melts, producing more dark areas, more heat absorption. Et cetera.
Another crucially important Earth systems factor here is that of currents, both in the air and in the oceans. Currents basically work on temperature gradients. If the temperature difference between two spots is steep, there’s movement, there’s exchange of water or air. If it’s shallow - for example because the oceans are heating up even deep down - the movement slows down. In the case of the big oceans currents (Gulf Stream & Co.), that spells huge trouble: if those slow down or even cease, there’d be catastrophic consequences. Think The Day After Tomorrow, not quite as fast as it happens in that film, but yes.
Likewise, atmospheric currents slow down when the temperature gradient becomes flatter, and then we end up with ‘stuck’ weather systems: they simply don’t move on. Which is why increased heat, bizarrely, creates long cold spells such as recent years’ Arctic vortexes.
As I said, this is just a very brief and simplified overview. On to…