Why do salt and sugar make ice melt?

Technically, they do not melt ice -- it requires heat to do that. But the addition of any solute (dissolved material) to water will lower its freezing point. So adding salt or sugar prevents melting ice from refreezing, or at least not until it gets much colder.

The solute has the effect of blocking molecules attempting to form lattices in the water (ice), so it requires a lower temperature before the same number of water molecules can form crystals.