Why does salt melt ice on a road?

Because salt water freezes below zero. Ice and liquid water are always exchanging molecules; dissolved salt gets in the way of molecules rejoining the ice but does nothing to stop them leaving it, so the balance tips towards melting and the freezing point drops - by around 20 degrees for a saturated brine.

That is why it stops working in extreme cold. Below about minus 10 rock salt is marginal, and below minus 18 it is useless, so colder places switch to calcium or magnesium chloride, or give up on melting and spread grit for traction instead. Salt also needs a little liquid to start with, which is why roads are usually treated before the snow rather than after.

The costs are real. Road salt corrodes cars and bridges, and it accumulates in soil and streams - chloride levels in some northern rivers have risen sharply over decades and do not go away. It is also how an ice cream churn worked before refrigeration: salt in the surrounding ice makes a brine well below zero, cold enough to freeze the mixture inside.