Short answers, pulled from the story.
Ice reflects far more solar energy back into space than open water or land cover. When higher temperatures decrease ice-covered area, the planet exposes more open water or land which decreases albedo and absorbs more solar energy leading to further warming.
Early climatologists such as Syukuro Manabe began making attempts to describe the role of ice cover in Earth's energy budget during the 1950s. By 1969 both USSR's Mikhail Ivanovich Budyko and the United States' William D. Sellers published papers presenting some of the first energy-balance climate models.
Recent Arctic sea ice decline is one of the primary factors behind the Arctic warming nearly four times faster than the global average since 1979. Continuous satellite readings of the Arctic sea ice began in that year and modelling studies show strong Arctic amplification only occurs when significant sea ice loss happens.
The end of the Snowball Earth periods involved the ice-albedo feedback again where deglaciation began once enough dust from erosion had built up in layers on the snow-ice surface to substantially lower its albedo. This likely started in the midlatitude regions and helped increase planet-wide temperature after glaciers near the equator formed about 717 million years ago.
Airborne particles such as black carbon and mineral dust are deposited on snow and ice surfaces causing a darkening effect that decreases albedo and accelerates melting. Microbial growth such as snow algae on glaciers and ice algae on sea ice can also cause a snow darkening effect which forms a positive feedback loop.