Why does Champagne become bubbly while an ordinary red wine made from fermenting grapes usually stays still?

Answer: Trapped second fermentation

Trapped second fermentation ✓ — Right: Champagne starts as a still base wine, then gets yeast and sugar in a sealed bottle. The yeast makes carbon dioxide, and because the bottle is closed, the gas dissolves under pressure instead of drifting away. Ordinary red wine also makes plenty of CO2 during fermentation, but most of it escapes from the vat or barrel before bottling.

More gas-rich grapes — Almost, but grapes are not born bubbly. Both red-wine grapes and Champagne grapes can provide sugar for yeast to ferment; the key difference is whether the carbon dioxide is deliberately trapped after the wine is already in a sealed bottle. Still red wines may retain a little dissolved CO2, yet winemakers usually keep it below the level that feels fizzy.

A stronger mushroom cork — The cork and wire cage hold pressure; they do not create it. A stronger closure is more like a lock on a steam boiler than the boiler itself. Without the sealed second fermentation producing CO2 first, a mushroom cork would simply be a fancy stopper on a still bottle.