Why does magma eventually burst out of a volcano instead of staying underground?

Answer: Gas-rich magma builds up pressure until it breaks through the rock above it

Gas-rich magma builds up pressure until it breaks through the rock above it ✓ — Correct! Magma is lighter than the solid rock around it, so it rises, and it carries dissolved gases such as water vapour and carbon dioxide. As it climbs and the pressure on it drops, those gases come out of solution and expand, like the fizz in a shaken bottle. The pressure in the chamber grows until it exceeds the strength of the overlying rock, and the magma forces its way out through cracks or the vent.

An earthquake shakes the magma loose and sets off the eruption — Wrong, mostly. Earthquakes and eruptions are both signs of a restless crust and often happen together, and a large quake can occasionally nudge a volcano that was already primed. But shaking does not create the pressure; the gas-charged magma does, and most eruptions come with no triggering quake at all.

Seawater seeps into the magma chamber and boils it over — Wrong. Magma chambers are sealed by kilometres of solid rock, and seawater does not trickle into them. Where water does meet magma near the surface it flashes to steam and can cause violent explosions, but the drive behind an eruption is the gas already dissolved in the magma, not an incoming flood.