Roads let your units move faster. That is what the "1/2 movement" thing refers to - for every point of movement a unit has, it can move 2 squares of road.
Roads, along with rivers and coasts (once you have researched Sailing) also connect your cities and resources to each other (known as your trade network), letting your cities gain the benefits of resources (such as health and happiness bonuses), and allowing your cities to have trade with each other (thereby generating income for your civ).
Similarly, you use roads to connect your civ's trade network to another civ's, thereby allowing your two civilizations to generate income from trade, and also allowing you to trade resources with that civ.