When the weight of the icecap increases due to accretion the glaciers do indeed move faster because the new ice is pushing them outward. When they push into the ocean they calve i.e. pieces, sometimes very large pieces, break off. The more ice is that is piling up in the interior the more is pushing into the ocean and the larger the chunks and the more often they are breaking off.
I know. The show was claiming that Greenland was losing more ice than it gained every year, and the accelerated glaciers were due to increased meltwater that accumulated under them and served as a sort of lubricant allowing the glaciers to “slide” faster (which would be true, though whether it’s actually happening in Greenland is debatable).