The mirror inside a vacuum flask
Blocking all three heat paths at once, and which one the silvering handles.
A vacuum flask keeps a drink hot by attacking conduction, convection and radiation separately, one part of its design for each, and the mirrored silvering on the glass is specifically there to stop the third of those, radiant heat, which the vacuum and the sealed lid alone cannot touch.
An empty gap stops two of the three
A vacuum flask is built from two glass walls with the air pumped out of the narrow gap between them, and that emptied gap is the first and most direct part of the design. Conduction and convection both need some material, a solid or a fluid, to carry heat, and a true vacuum contains neither. Heat trying to conduct outward from the hot inner wall to the cooler outer one finds nothing in the gap to pass through, and heat trying to convect away finds no air to circulate and carry it. That alone deals with two of the three transport mechanisms almost completely.
The sealed lid handles a separate problem, stopping the warmed air above the liquid from rising out through an open neck and being replaced by cold air rushing in, the same convective loss any open hot drink suffers. That makes three weaknesses and three answers, each aimed at one route heat can travel by and largely indifferent to the other two.
What the silvering does to radiant heat
Radiation needs no material at all. It travels across an empty vacuum as easily as across open air, since it is simply energy given off by anything warm toward anything cooler nearby, and the vacuum gap does nothing to stop it. This is where the mirrored silvering matters: a thin reflective coating on the glass surfaces facing into the gap, which sends most of the radiant heat trying to cross back toward the hot liquid instead of letting the outer wall absorb it. The effect is large. Bare glass gives off radiant heat at close to the most any surface at its temperature can, while a bright silver surface gives off only a few percent of that and reflects nearly everything that lands on it.
A shiny foil sun-shade propped inside a car's windscreen on a sunny day keeps the car cooler than a plain dark shade of the same thickness, even though both are opaque and both keep the sun off the seats. The difference lies in what each shade does with the radiant heat that reaches it. A dark shade absorbs that heat readily and then gives a good share of it off again inward, warming the air trapped behind it, while a shiny foil shade reflects much of the incoming heat straight back out through the glass before it can be absorbed. The mirrored inside of a flask does the same job in miniature and in the opposite direction, turning heat escaping from the hot liquid back inward before the glass can soak it up and pass it on to the air outside.
Attacking all three routes at once is what lets a good flask keep a drink hot from breakfast until the afternoon, where an open cup in the same room would be lukewarm within the hour.
Which defence has failed
Even an undamaged flask has one route it cannot close. The inner and outer walls have to meet somewhere, and they meet at the neck, where a ring of solid glass or steel joins them and a stopper sits in the opening. Heat conducts out through that ring and through the stopper, and in a good flask this small region is where most of the remaining heat escapes, which is why stoppers are made thick and from poorly conducting plastic or cork.
Seeing the flask as three separate defences explains why a damaged one can fail in such different ways. A flask with a cracked or leaking vacuum seal loses most of its performance almost at once, since air getting back into the gap restores both conduction and convection, even though the silvering and the lid are still intact. A flask with scratched or tarnished silvering keeps working reasonably well in the short term, since the vacuum is still stopping conduction and convection, but it loses heat faster than a fresh one over longer periods, because radiant heat now has an easier path across the gap. A flask left with its lid off loses heat mainly by letting warmed air escape upward and cold air take its place, a route that bypasses the double-walled body entirely.
A flask filled only partway, leaving a large pocket of air above the liquid inside the sealed inner chamber, also loses heat somewhat faster than one filled close to the top, because that pocket is free to convect within the flask's own inner space, carrying warmth up toward the lid and neck while every other route is sealed. And every opening a drink can be poured from is a gap in the sealed system, so a flask poured from repeatedly in quick succession loses heat through its mouth however good its vacuum or silvering.